รบกวนตรวจด้วยครับ
2. $\int \frac{dx}{1-cos3x} $
$= \int \frac{1+cos3x}{sin^23x}dx $
$= \frac{1}{3}\int cosec^23xd3x-\frac{1}{3}\int \frac{1}{sin^23x}dsin3x $
$= \frac{-cot^23x+cosec3x}{3}++c$
3. $\int (sec^24x-1)^2dx$
= $\frac{1}{4}\int sec^24x d4x-\frac{1}{2}\int sec4x d4x +x+c$
= $\frac{tan4x}{4}-\frac{1}{2}ln|sec4x+tan4x|+x+c$
= $\frac{tan4x}{4}+\frac{1}{2}ln|sec4x-tan4x|+x+c$
4. $\int sin7xcos5x dx $
= $\int sin12x+sin2x dx$
= $-\frac{cos12x}{24}-\frac{cos2x}{4}+c $
5. $\int \frac{e^x}{\sqrt{1-e^2x}} dx$
= $\int \frac{1}{\sqrt{1-e^2x}} de^x$ ให้ $e^x = sin\theta$
= $\int \frac{1}{cos\theta} dsin\theta$
= $-\int \frac{1}{cos\theta} dsin(\frac{\pi}{2}-\theta)$
= $-ln|cos\theta|+c$
= $ln|sec\theta|+c$
6. $\int \frac{x^2}{\sqrt{4-x^2}} dx$
ให้ $x=2cos\theta$
= $-\int \frac{4(1-2sin^2\theta)}{2sin\theta} d2(cos\frac{\pi}{2}-\theta)$
= $-4\int (sin\theta)^{-2}dsin\theta - 8sin\theta+c$
= $4cosec\theta-4sin\theta+c$
= $\frac{16}{\sqrt{4-x^2}}-4\sqrt{4-x^2}+c$