Question

The steps to integrate this problem using trigonometric substitution.

Answer 1

`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`