The steps to integrate this problem using trigonometric substitution.

Question 1

Question 2

Answer:

I=(x^3)/(3(25x^2+1)^(3/2))

Explanation:

(a)

$5x=tan{\theta}\\x = (tan(\theta))/(5)\\dx=(1)/(5)\sec^2{\theta}d\theta\\$

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

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

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

$I=(1)/(125)\int\sin^2{\theta}{cos{\theta}}d\theta$

$I=(1)/(125)\int\sin^2{\theta}dsin(\theta)$

$I=(1)/(125)(1)/(3)sin^3{\theta}}$

I=(1)/(125)(1)/(3)((5x)/(√(25x^2+1))))^3

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