2014年10月18日 · There is an easy way to evaluate $\int \sin^n(x)\cos^m(x)\,dx$ when at least one of $n,m$ is odd. Say $n$ is odd. Then keep one factor of $\sin$ and convert all the rest to $\cos$ , and substitute $u=\cos x, du=-\sin x\,dx$ .

Here's my attempt at the problem: $\int{\tan^{5}(x)\sec^4(x)}dx= \int{\frac{\sin^5(x)}{\cos^9(x)}}\,dx= \int{\frac{\sin(x)(\sin^2(x))^2}{\cos^9(x)}}\,dx= \int{\frac{\sin(x)(1-\cos^2(x))^2}{\cos^9(...

曾答过被积分是 \sin^n x, 这里 \int \tan ^nx\mathrm dx 也可以有递推式： \begin{align*} I_n=\int\tan^nx\mathrm dx &=\int\tan^{n-2}(\tan^2x+1-1)\mathrm dx\\[1em] &=\int\tan^{n-2}\sec^2x\mathrm dx-I_{n-2}\\[1em] &=\int\tan^{n-2}\mathrm d\tan x-I_{n-2}\\[1em] &=\frac {1}{n-1}\tan^{n-1}x-I_{n-2},{~}n\geqslant 2\end{align*}\\

\int e^x\cos (x)dx \int_{0}^{\pi}\sin(x)dx \sum_{n=0}^{\infty}\frac{3}{2^n} Show More

2023年12月10日 · 一、tanx类1. tanx 正切、 cotx=\frac{1}{tanx} 余切、 secx=\frac{1}{cos} 正割2. arctan^{'}x=\frac{1}{1+x^{2}} , arccot^{'}x=-\frac{1}{1+x^{2}} 3. \int_{}^{}d(tanx)=\int_{}^{}sec^{2}xdx ( tan^{…

2023年11月10日 · 先给出 三角函数 常用基础积分表及 导数 表. 1. \int\tan x\mathrm {d}x=-\ln\mid \cos x\mid+C; (\tan x)'=\sec^2x.\\ 解 \quad \begin {align*}\int\tan x\mathrm {d}x&=\int\frac {\sin x} {\cos x}\mathrm {d}x\\&=-\int\frac {1} {\cos x}\mathrm {d}\cos x\\&=-\ln\mid\cos x\mid+C.\end {align*}\\ 2. \int \cot x\mathrm {d}x=\ln\mid\sin x\mid+C; (\cot x)'=-\csc^2x.\\

使用 递推公式 \int tan^ {n}xdx=\frac {tan^ {n-1}x} {n-1}-\int tan^ {n-2}xdx. 取 n=3 ，有. \int tan^3xdx=\frac {tan^2x} {2}-\int tanxdx=\frac {tan^2x} {2}+ln|cosx|+C.

$$=\displaystyle \int (tan^3xsec^2x - tanx(sec^2x-1))dx$$ $$=\displaystyle \int (tan^3xsec^2x - tanxsec^2x + tanx)dx$$ $$=\dfrac{tan^4x}{4} - \dfrac{tan^2x}{2} + ln|secx| + C$$

\int\frac{1}{a\sin x+b\cos x}\text{d}x=\frac{1}{\sqrt{a^2+b^2}}\ln\left|\tan\frac{x+\arctan\frac{b}{a}}{2}\right|+C 被积函数有 \sqrt{a^2-x^2} , 可令 x=a\sin t 被积函数有 \sqrt{a^2+x^2} , 可令 x=a\tan t

Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music…