Question

The steps to integrate this problem using trigonometric substitution.

Answer 1

Answer:

$I=\frac{x^3}{3(25x^2+1)^{3/2}}$

Step-by-step explanation:

(a)

$5x=tan{\theta}\\x = \frac{\tan{\theta}}{5}\\dx=\frac{1}{5}\sec^2{\theta}d\theta$

$I=\frac{1}{125}\int\frac{tan^2{\theta}sec^2{\theta}}{(tan^2{\theta}+1)^{5/2}}d\theta$

$I=\frac{1}{125}\int\frac{tan^2{\theta}sec^2{\theta}}{sec^5{\theta}}d\theta$

$I=\frac{1}{125}\int\frac{tan^2{\theta}}{sec^3{\theta}}d\theta$

$I=\frac{1}{125}\int\sin^2{\theta}{cos{\theta}}d\theta$

$I=\frac{1}{125}\int\sin^2{\theta}d\sin{\theta}$

$I=\frac{1}{125}\frac{1}{3}sin^3{\theta}}$

$I=\frac{1}{125}\frac{1}{3}(\frac{5x}{\sqrt{25x^2+1})})^3$