MIT_单变量微积分_24

版权声明： https://blog.csdn.net/qq_38386316/article/details/88805652

1.三角函数的积分以及三角替换

三角学常用函数：

• $sin^2\Theta +cos^2\Theta=1$
• $cos(2\Theta)=cos^2\Theta-sin^2\Theta$
• $sin(2\Theta)=2sin\Theta cos \Theta$

半角公式：
$cos(2\Theta)=cos^2\Theta-sin^2\Theta\\ =cos^2\Theta-(1-cos^2\Theta)\\ =2cos^2(2\Theta)-1$
可以得知：
$cos^2\Theta=\frac{1+cos(2\Theta)}{2}\\ sin^2\Theta=\frac{1-cos(2\Theta)}{2}\\ dsinx=(cosx)dx\Rightarrow \int cosx dx=sinx+C\\ dcosx=(-sinx)dx\Rightarrow \int sinx dx=-cosx+C\\$
Ex: $\int sin^n(x)cos^m(x)dx(m,n=0,1,2.....)$m,n至少有一个奇数.
(1) $m=1$
令 $u=sinx$,则：
$\int sin^n(x)cosxdx=\int u^n dx\\ =\int u^n du\\ =\frac{u^{n+1}}{n+1}+C\\ =\frac{(sinx)^{n+1}}{n+1}+C$
(2) $n=3,m=2$
$\int sin^3x cos^2x dx\\ =\int(1-cos ^2x)sinx cos^2xdx\\ =\int(cos^2x-cos^4x)sinxdx\\ 令u=cosx,du=-sinxdx\\ =\int(u^2-u^4)(-du)\\ =\frac{1}{5}u^5-\frac{1}{3}u^3+C\\ =\frac{1}{5}(cosx)^5-\frac{1}{3}(cosx)^3+C$
(3) $n=3,m=0$
$\int sin^3xdx\\ =\int(1-cos^2x)sinxdx\\ 令u=cosx,du=-sinxdx\\ =\int(1-u^2)x(-du)\\ =\frac{1}{3}u^3-u+C\\ =\frac{1}{3}(cosx)^3-cosx+C$

半角公式的使用：
$\int cos^2xdx=\int \frac{1+cos(2x)}{2}dx\\ =\frac{1}{2}x+\frac{sin(2x)}{4}+C$
Ex:
$\int sin^2cos^2dx\\ =\int(\frac{1-cos(2x)}{2})(\frac{1+cos(2x)}{2})dx\\ =\int\frac{1-cos^2(2x)}{4}dx\\ =\int(\frac{1}{4}-\frac{1+cos(4x)}{4*2})dx\\ =\int(\frac{1}{8}-\frac{cos4x}{8})dx\\ =\frac{1}{8}x-\frac{sin4x}{8}+C$
Ex
$sin^2xcos^2x=(sinxcosx)^2\\ =(\frac{sin(2x)}{2})^2\\ =\frac{sin^2(2x)}{4}\\ =\frac{1}{4}(\frac{1-cos(4x)}{2})$
三角替换：
$\int \sqrt{a^2-y^2}dy\\ =\int(acos\Theta)(acos\Theta d\Theta)\\ =a^2\int cos^2 \Theta d\Theta\\ =a^2 (\frac{\Theta}{2}+\frac{sin(2\Theta)}{4})+C$