✨Danh sách tích phân với hàm lượng giác

Đây là danh sách tích phân (nguyên hàm) của các hàm lượng giác. Đối với tích phân của chứa hàm lượng giác và hàm mũ, xem Danh sách tích phân với hàm mũ. Đối với danh sách đầy đủ các tích phân, xem Danh sách tích phân. Đối với danh sách các tích phân đặc biệt của các hàm lượng giác, xem Tích phân lượng giác.

Nhìn chung, với $\backslash cos(x)$ là đạo hàm của hàm số $\backslash sin(x)$, ta có

: $\backslash int\; a\backslash cos\; nx\backslash ,dx\; =\; \backslash frac\{a\}\{n\}\backslash sin\; nx+C$ Trong mọi công thức dưới đây, là một hằng số khác không và ký hiệu cho hằng số tích phân.

Tích phân chỉ chứa hàm sin

: $\backslash int\backslash sin\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a\}\backslash cos\; ax+C$

: $\backslash int\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{4a\}\; \backslash sin\; 2ax\; +C=\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\; \backslash sin\; ax\backslash cos\; ax\; +C$

: $\backslash int\backslash sin^3\; \{ax\}\backslash ,dx\; =\; \backslash frac\{\backslash cos\; 3ax\}\{12a\}\; -\; \backslash frac\{3\; \backslash cos\; ax\}\{4a\}\; +C$

: $\backslash int\; x\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^2\}\{4\}\; -\; \backslash frac\{x\}\{4a\}\; \backslash sin\; 2ax\; -\; \backslash frac\{1\}\{8a^2\}\; \backslash cos\; 2ax\; +C$

: $\backslash int\; x^2\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{6\}\; -\; \backslash left(\; \backslash frac\; \{x^2\}\{4a\}\; -\; \backslash frac\{1\}\{8a^3\}\; \backslash right)\; \backslash sin\; 2ax\; -\; \backslash frac\{x\}\{4a^2\}\; \backslash cos\; 2ax\; +C$

: $\backslash int\; x\backslash sin\; ax\backslash ,dx\; =\; \backslash frac\{\backslash sin\; ax\}\{a^2\}-\backslash frac\{x\backslash cos\; ax\}\{a\}+C$

: $\backslash int(\backslash sin\; b\_1x)(\backslash sin\; b\_2x)\backslash ,dx\; =\; \backslash frac\{\backslash sin((b\_2-b\_1)x)\}\{2(b\_2-b\_1)\}-\backslash frac\{\backslash sin((b\_1+b\_2)x)\}\{2(b\_1+b\_2)\}+C\; \backslash qquad\backslash mbox\{(\}|b\_1|\backslash neq|b\_2|\backslash mbox\{)\}$

: $\backslash int\backslash sin^n\; \{ax\}\backslash ,dx\; =\; -\backslash frac\{\backslash sin^\{n-1\}\; ax\backslash cos\; ax\}\{na\}\; +\; \backslash frac\{n-1\}\{n\}\backslash int\backslash sin^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(\}n>0\backslash mbox\{)\}$

: $\backslash int\backslash frac\{dx\}\{\backslash sin\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}+\backslash cot\{ax\}\backslash right|\}+C$

: $\backslash int\backslash frac\{dx\}\{\backslash sin^n\; ax\}\; =\; \backslash frac\{\backslash cos\; ax\}\{a(1-n)\; \backslash sin^\{n-1\}\; ax\}+\backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash sin^\{n-2\}ax\}\; \backslash qquad\backslash mbox\{(\}n>1\backslash mbox\{)\}$

: {k=0}^{2k\leq n} (-1)^{k+1} \frac{x^{n-2k{a^{1+2k\frac{n!}{(n-2k)!} \cos ax +\sum{k=0}^{2k+1\leq n}(-1)^k \frac{x^{n-1-2k{a^{2+2k\frac{n!}{(n-2k-1)!} \sin ax \ &= - \sum_{k=0}^n \frac{x^{n-k{a^{1+k\frac{n!}{(n-k)!}\cos\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(}n>0\mbox{)} \end{align}

: $\backslash int\backslash frac\{\backslash sin\; ax\}\{x\}\backslash ,dx\; =\; \backslash sum\_\{n=0\}^\backslash infty\; (-1)^n\backslash frac\{(ax)^\{2n+1\{(2n+1)\backslash cdot\; (2n+1)!\}\; +C$

: $\backslash int\backslash frac\{\backslash sin\; ax\}\{x^n\}\backslash ,dx\; =\; -\backslash frac\{\backslash sin\; ax\}\{(n-1)x^\{n-1\; +\; \backslash frac\{a\}\{n-1\}\backslash int\backslash frac\{\backslash cos\; ax\}\{x^\{n-1\backslash ,dx$ : $\backslash int\backslash frac\{dx\}\{1\backslash pm\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\backslash mp\backslash frac\{\backslash pi\}\{4\}\backslash right)+C$

: $\backslash int\backslash frac\{x\backslash ,dx\}\{1+\backslash sin\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\; -\; \backslash frac\{\backslash pi\}\{4\}\backslash right)+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash cos\backslash left(\backslash frac\{ax\}\{2\}-\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$

: $\backslash int\backslash frac\{x\backslash ,dx\}\{1-\backslash sin\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash cot\backslash left(\backslash frac\{\backslash pi\}\{4\}\; -\; \backslash frac\{ax\}\{2\}\backslash right)+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{4\}-\backslash frac\{ax\}\{2\}\backslash right)\backslash right|+C$

: $\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{1\backslash pm\backslash sin\; ax\}\; =\; \backslash pm\; x+\backslash frac\{1\}\{a\}\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}\backslash mp\backslash frac\{ax\}\{2\}\backslash right)+C$

Tích phân chỉ chứa hàm cos

: $\backslash int\backslash cos\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a\}\backslash sin\; ax+C$

: $\backslash int\backslash cos^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{4a\}\; \backslash sin\; 2ax\; +C\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\; \backslash sin\; ax\backslash cos\; ax\; +C$

: $\backslash int\backslash cos^n\; ax\backslash ,dx\; =\; \backslash frac\{\backslash cos^\{n-1\}\; ax\backslash sin\; ax\}\{na\}\; +\; \backslash frac\{n-1\}\{n\}\backslash int\backslash cos^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(\}n>0\backslash mbox\{)\}$

: $\backslash int\; x\backslash cos\; ax\backslash ,dx\; =\; \backslash frac\{\backslash cos\; ax\}\{a^2\}\; +\; \backslash frac\{x\backslash sin\; ax\}\{a\}+C$

: $\backslash int\; x^2\backslash cos^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{6\}\; +\; \backslash left(\; \backslash frac\; \{x^2\}\{4a\}\; -\; \backslash frac\{1\}\{8a^3\}\; \backslash right)\; \backslash sin\; 2ax\; +\; \backslash frac\{x\}\{4a^2\}\; \backslash cos\; 2ax\; +C$

: {k=0}^{2k+1\leq n} (-1)^{k} \frac{x^{n-2k-1{a^{2+2k\frac{n!}{(n-2k-1)!} \cos ax +\sum{k=0}^{2k\leq n}(-1)^{k} \frac{x^{n-2k{a^{1+2k\frac{n!}{(n-2k)!} \sin ax \ &=\sum{k=0}^n (-1)^{\lfloor k/2 \rfloor} \frac{x^{n-k{a^{1+k\frac{n!}{(n-k)!}\cos\left(ax -\frac{(-1)^k+1}{2}\frac{\pi}{2}\right) \ &=\sum{k=0}^n \frac{x^{n-k{a^{1+k\frac{n!}{(n-k)!}\sin\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(}n>0\mbox{)} \end{align}

: $\backslash int\backslash frac\{\backslash cos\; ax\}\{x\}\backslash ,dx\; =\; \backslash ln|ax|+\backslash sum\_\{k=1\}^\backslash infty\; (-1)^k\backslash frac\{(ax)^\{2k\{2k\backslash cdot(2k)!\}+C$

: $\backslash int\backslash frac\{\backslash cos\; ax\}\{x^n\}\backslash ,dx\; =\; -\backslash frac\{\backslash cos\; ax\}\{(n-1)x^\{n-1-\backslash frac\{a\}\{n-1\}\backslash int\backslash frac\{\backslash sin\; ax\}\{x^\{n-1\backslash ,dx\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{dx\}\{\backslash cos\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$

: $\backslash int\backslash frac\{dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{\backslash sin\; ax\}\{a(n-1)\; \backslash cos^\{n-1\}\; ax\}\; +\; \backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash cos^\{n-2\}\; ax\}\; \backslash qquad\backslash mbox\{(\}n>1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{dx\}\{1+\backslash cos\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}+C$

: $\backslash int\backslash frac\{dx\}\{1-\backslash cos\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+C$

: $\backslash int\backslash frac\{x\backslash ,dx\}\{1+\backslash cos\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}\; +\; \backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash cos\backslash frac\{ax\}\{2\}\backslash right|+C$

: $\backslash int\backslash frac\{x\backslash ,dx\}\{1-\backslash cos\; ax\}\; =\; -\backslash frac\{x\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash sin\backslash frac\{ax\}\{2\}\backslash right|+C$

: $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{1+\backslash cos\; ax\}\; =\; x\; -\; \backslash frac\{1\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}+C$

: $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{1-\backslash cos\; ax\}\; =\; -x-\backslash frac\{1\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+C$

: $\backslash int(\backslash cos\; a\_1x)(\backslash cos\; a\_2x)\backslash ,dx\; =\; \backslash frac\{\backslash sin((a\_2-a\_1)x)\}\{2(a\_2-a\_1)\}+\backslash frac\{\backslash sin((a\_2+a\_1)x)\}\{2(a\_2+a\_1)\}+C\; \backslash qquad\backslash mbox\{(\}|a\_1|\backslash neq|a\_2|\backslash mbox\{)\}$

Tích phân chỉ chứa hàm tan

: $\backslash int\backslash tan\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a\}\backslash ln|\backslash cos\; ax|+C\; =\; \backslash frac\{1\}\{a\}\backslash ln|\backslash sec\; ax|+C\backslash ,!$

: $\backslash int\; \backslash tan^2\{x\}\; \backslash ,\; dx\; =\; \backslash tan\{x\}\; -\; x\; +C$

: $\backslash int\backslash tan^n\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a(n-1)\}\backslash tan^\{n-1\}\; ax-\backslash int\backslash tan^\{n-2\}\; ax\backslash ,dx\; \backslash qquad(n\backslash neq\; 1)\backslash ,!$

: $\backslash int\backslash frac\{dx\}\{q\; \backslash tan\; ax\; +\; p\}\; =\; \backslash frac\{1\}\{p^2\; +\; q^2\}(px\; +\; \backslash frac\{q\}\{a\}\backslash ln|q\backslash sin\; ax\; +\; p\backslash cos\; ax|)+C\; \backslash qquad(p^2\; +\; q^2\backslash neq\; 0)\backslash ,!$

: $\backslash int\backslash frac\{dx\}\{\backslash tan\; ax\; +\; 1\}\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; +\; \backslash cos\; ax|+C\backslash ,!$

: $\backslash int\backslash frac\{dx\}\{\backslash tan\; ax\; -\; 1\}\; =\; -\backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; -\; \backslash cos\; ax|+C\backslash ,!$

: $\backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax\; +\; 1\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; +\; \backslash cos\; ax|+C\backslash ,!$

: $\backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax\; -\; 1\}\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; -\; \backslash cos\; ax|+C\backslash ,!$

Tích phân chỉ chứa hàm secant

:$\backslash int\; \backslash sec\{ax\}\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash sec\{ax\}\; +\; \backslash tan\{ax\}\backslash right|\}+C$

:$\backslash int\; \backslash sec^2\{x\}\; \backslash ,\; dx\; =\; \backslash tan\{x\}+C$

:$\backslash int\; \backslash sec^n\{ax\}\; \backslash ,\; dx\; =\; \backslash frac\{\backslash sec^\{n-2\}\{ax\}\; \backslash tan\; \{ax\{a(n-1)\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash sec^\{n-2\}\{ax\}\; \backslash ,\; dx\; \backslash qquad\; \backslash mbox\{(\}n\backslash ne\; 1\backslash mbox\{)\}\backslash ,!$

:$\backslash int\; \backslash sec^n\{x\}\; \backslash ,\; dx\; =\; \backslash frac\{\backslash sec^\{n-2\}\{x\}\backslash tan\{x\{n-1\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash sec^\{n-2\}\{x\}\backslash ,dx$

:$\backslash int\; \backslash frac\{dx\}\{\backslash sec\{x\}\; +\; 1\}\; =\; x\; -\; \backslash tan\{\backslash frac\{x\}\{2+C$

:$\backslash int\; \backslash frac\{dx\}\{\backslash sec\{x\}\; -\; 1\}\; =\; -\; x\; -\; \backslash cot\{\backslash frac\{x\}\{2+C$

Tích phân chỉ chứa hàm cosecant

:

:$\backslash int\; \backslash csc^2\{x\}\; \backslash ,\; dx\; =\; -\backslash cot\{x\}+C$

:

:$\backslash int\; \backslash csc^n\{ax\}\; \backslash ,\; dx\; =\; -\backslash frac\{\backslash csc^\{n-2\}\{ax\}\; \backslash cot\{ax\{a(n-1)\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash csc^\{n-2\}\{ax\}\; \backslash ,\; dx\; \backslash qquad\; \backslash mbox\{\; (\}n\; \backslash ne\; 1\backslash mbox\{)\}$

:$\backslash int\; \backslash frac\{dx\}\{\backslash csc\{x\}\; +\; 1\}\; =\; x\; -\; \backslash frac\{2\}\{\backslash cot\{\backslash frac\{x\}\{2+1\}+C$

:$\backslash int\; \backslash frac\{dx\}\{\backslash csc\{x\}\; -\; 1\}\; =\; -\; x\; +\; \backslash frac\{2\}\{\backslash cot\{\backslash frac\{x\}\{2-1\}+C$

Tích phân chỉ chứa hàm cotang

:$\backslash int\backslash cot\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a\}\backslash ln|\backslash sin\; ax|+C$

:$\backslash int\; \backslash cot^2\{x\}\; \backslash ,\; dx\; =\; -\backslash cot\{x\}\; -\; x\; +C$

: $\backslash int\backslash cot^n\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a(n-1)\}\backslash cot^\{n-1\}\; ax\; -\; \backslash int\backslash cot^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{dx\}\{1\; +\; \backslash cot\; ax\}\; =\; \backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax+1\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; +\; \backslash cos\; ax|+C$

: $\backslash int\backslash frac\{dx\}\{1\; -\; \backslash cot\; ax\}\; =\; \backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax-1\}\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; -\; \backslash cos\; ax|+C$

Tích phân chứa hàm sin và cos

Tích phân một hàm hữu tỉ (phân thức) của và có thể được tính bằng quy tắc Bioche.

: $\backslash int\backslash frac\{dx\}\{\backslash cos\; ax\backslash pm\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\backslash sqrt\{2\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\backslash pm\backslash frac\{\backslash pi\}\{8\}\backslash right)\backslash right|+C$

: $\backslash int\backslash frac\{dx\}\{(\backslash cos\; ax\backslash pm\backslash sin\; ax)^2\}\; =\; \backslash frac\{1\}\{2a\}\backslash tan\backslash left(ax\backslash mp\backslash frac\{\backslash pi\}\{4\}\backslash right)+C$

: $\backslash int\backslash frac\{dx\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^n\}\; =\; \backslash frac\{1\}\{n-1\}\backslash left(\backslash frac\{\backslash sin\; x\; -\; \backslash cos\; x\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^\{n\; -\; 1\; -\; 2(n\; -\; 2)\backslash int\backslash frac\{dx\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^\{n-2\; \backslash right)$

: $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{\backslash cos\; ax\; \backslash pm\; \backslash sin\; ax\}\; =\; \backslash frac\{x\}\{2\}\; \backslash pm\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; \backslash pm\; \backslash cos\; ax\backslash right|+C$

: $\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{\backslash cos\; ax\; \backslash pm\; \backslash sin\; ax\}\; =\; \backslash pm\backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; \backslash pm\; \backslash cos\; ax\backslash right|+C$

: $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{(\backslash sin\; ax)(1+\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{4a\}\backslash tan^2\backslash frac\{ax\}\{2\}+\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|+C$

: $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{(\backslash sin\; ax)(1-\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{4a\}\backslash cot^2\backslash frac\{ax\}\{2\}-\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|+C$

: $\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{(\backslash cos\; ax)(1+\backslash sin\; ax)\}\; =\; \backslash frac\{1\}\{4a\}\backslash cot^2\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)+\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$

: $\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{(\backslash cos\; ax)(1-\backslash sin\; ax)\}\; =\; \backslash frac\{1\}\{4a\}\backslash tan^2\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)-\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$

: $\backslash int(\backslash sin\; ax)(\backslash cos\; ax)\backslash ,dx\; =\; \backslash frac\{1\}\{2a\}\backslash sin^2\; ax\; +C$

: $\backslash int(\backslash sin\; a\_1x)(\backslash cos\; a\_2x)\backslash ,dx\; =\; -\backslash frac\{\backslash cos((a\_1-a\_2)x)\}\{2(a\_1-a\_2)\}\; -\backslash frac\{\backslash cos((a\_1+a\_2)x)\}\{2(a\_1+a\_2)\}\; +C\backslash qquad\backslash mbox\{(\}|a\_1|\backslash neq|a\_2|\backslash mbox\{)\}$

: $\backslash int(\backslash sin^n\; ax)(\backslash cos\; ax)\backslash ,dx\; =\; \backslash frac\{1\}\{a(n+1)\}\backslash sin^\{n+1\}\; ax\; +C\backslash qquad\backslash mbox\{(\}n\backslash neq\; -1\backslash mbox\{)\}$

: $\backslash int(\backslash sin\; ax)(\backslash cos^n\; ax)\backslash ,dx\; =\; -\backslash frac\{1\}\{a(n+1)\}\backslash cos^\{n+1\}\; ax\; +C\backslash qquad\backslash mbox\{(\}n\backslash neq\; -1\backslash mbox\{)\}$

:

: $\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos\; ax)\}\; =\; \backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\; ax\backslash right|+C$

: $\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos^n\; ax)\}\; =\; \backslash frac\{1\}\{a(n-1)\backslash cos^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos^\{n-2\}\; ax)\}\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{dx\}\{(\backslash sin^n\; ax)(\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{a(n-1)\backslash sin^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{(\backslash sin^\{n-2\}\; ax)(\backslash cos\; ax)\}\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{1\}\{a(n-1)\backslash cos^\{n-1\}\; ax\}\; +C\backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{\backslash sin^2\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash sin\; ax+\backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}+\backslash frac\{ax\}\{2\}\backslash right)\backslash right|+C$

: $\backslash int\backslash frac\{\backslash sin^2\; ax\backslash ,dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{\backslash sin\; ax\}\{a(n-1)\backslash cos^\{n-1\}ax\}-\backslash frac\{1\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash cos^\{n-2\}ax\}\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{\backslash sin^n\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; =\; -\backslash frac\{\backslash sin^\{n-1\}\; ax\}\{a(n-1)\}\; +\; \backslash int\backslash frac\{\backslash sin^\{n-2\}\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

:

: $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{\backslash sin^n\; ax\}\; =\; -\backslash frac\{1\}\{a(n-1)\backslash sin^\{n-1\}\; ax\}\; +C\backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{\backslash cos^2\; ax\backslash ,dx\}\{\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash left(\backslash cos\; ax+\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|\backslash right)\; +C$

: $\backslash int\backslash frac\{\backslash cos^2\; ax\backslash ,dx\}\{\backslash sin^n\; ax\}\; =\; -\backslash frac\{1\}\{n-1\}\backslash left(\backslash frac\{\backslash cos\; ax\}\{a\backslash sin^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{\backslash sin^\{n-2\}\; ax\}\backslash right)\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

:

Tích phân chứa hàm sin và tang

: $\backslash int\; \backslash sin\; ax\; \backslash tan\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a\}(\backslash ln|\backslash sec\; ax\; +\; \backslash tan\; ax|\; -\; \backslash sin\; ax)+C\backslash ,!$

: $\backslash int\backslash frac\{\backslash tan^n\; ax\backslash ,dx\}\{\backslash sin^2\; ax\}\; =\; \backslash frac\{1\}\{a(n-1)\}\backslash tan^\{n-1\}\; (ax)\; +C\backslash qquad(n\backslash neq\; 1)\backslash ,!$

Tích phân chứa hàm cos và tang

: $\backslash int\backslash frac\{\backslash tan^n\; ax\backslash ,dx\}\{\backslash cos^2\; ax\}\; =\; \backslash frac\{1\}\{a(n+1)\}\backslash tan^\{n+1\}\; ax\; +C\backslash qquad(n\backslash neq\; -1)\backslash ,!$

Tích phân chứa hàm sin và cotang

: $\backslash int\backslash frac\{\backslash cot^n\; ax\backslash ,dx\}\{\backslash sin^2\; ax\}\; =\; -\backslash frac\{1\}\{a(n+1)\}\backslash cot^\{n+1\}\; ax\; +C\backslash qquad(n\backslash neq\; -1)\backslash ,!$

Tích phân chứa hàm cos và cotang

: $\backslash int\backslash frac\{\backslash cot^n\; ax\backslash ,dx\}\{\backslash cos^2\; ax\}\; =\; \backslash frac\{1\}\{a(1-n)\}\backslash tan^\{1-n\}\; ax\; +C\backslash qquad(n\backslash neq\; 1)\backslash ,!$

Tích phân chứa hàm secant và tang

: $\backslash int(\backslash sec\; x)(\backslash tan\; x)\backslash ,dx=\; \backslash sec\; x\; +\; C$

Tích phân chứa hàm cosecant và cotang

: $\backslash int(\backslash csc\; x)(\backslash cot\; x)\backslash ,dx=\; -\backslash csc\; x\; +\; C$

Tích phân trên một phần tư đường tròn

: $\backslash int$^} \sin^n x \, dx = \int^} \cos^n x \, dx = \begin{cases} \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}, & n=2,4,6,8,\ldots \ \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{4}{5} \cdot \frac{2}{3}, & n=3,5,7,9,\ldots \ 1, & n=1 \end{cases}

Tích phân với giới hạn đối xứng

: $\backslash int\_^\backslash sin\{x\}\backslash ,dx\; =\; 0$

: $\backslash int$^\cos {x}\,dx = 2\int^\cos {x}\,dx = 2\int_^\cos {x}\,dx = 2\sin {c}

: $\backslash int\_^\backslash tan\; \{x\}\backslash ,dx\; =\; 0$

: $\backslash int\_\{-\backslash frac\{a\}\{2^\{\backslash frac\{a\}\{2\; x^2\backslash cos^2\; \{\backslash frac\{n\backslash pi\; x\}\{a\backslash ,dx\; =\; \backslash frac\{a^3(n^2\backslash pi^2-6)\}\{24n^2\backslash pi^2\}\; \backslash qquad$ ( là số nguyên dương lẻ)

: $\backslash int\_\{\backslash frac\{-a\}\{2^\{\backslash frac\{a\}\{2\; x^2\backslash sin^2\; \{\backslash frac\{n\backslash pi\; x\}\{a\backslash ,dx\; =\; \backslash frac\{a^3(n^2\backslash pi^2-6(-1)^n)\}\{24n^2\backslash pi^2\}\; =\; \backslash frac\{a^3\}\{24\}\; (1-6\backslash frac\{(-1)^n\}\{n^2\backslash pi^2\})\; \backslash qquad$ ( là số nguyên dương)

Tích phân trên toàn bộ đường tròn

: $\backslash int\_^\backslash sin^\{2m+1\}\{x\}\backslash cos^\{2n+1\}\{x\}\backslash ,dx\; =\; 0\; \backslash qquad\; m,n\; \backslash in\; \backslash mathbb\{Z\}$