# Integration by Trig substitution

• Feb 11th 2010, 02:56 AM
racewithferrari
Integration by Trig substitution
Find the integration of :
(x^3)/(sqrt(x^2+100))

I did the following work, but it seems my answer is not matching to the Wolfram|Alpha' answer.
Attachment 15355

int x^3/(sqrt( x^2 +100)) dx - Wolfram|Alpha
• Feb 11th 2010, 03:24 AM
dedust
Quote:

Originally Posted by racewithferrari
Find the integration of :
(x^3)/(sqrt(x^2+100))

I did the following work, but it seems my answer is not matching to the Wolfram|Alpha' answer.
Attachment 15355

int x^3/(sqrt( x^2 +100)) dx - Wolfram|Alpha

it is the same,..just in different form,..try to simplify it
• Feb 11th 2010, 03:32 AM
Prove It
Quote:

Originally Posted by racewithferrari
Find the integration of :
(x^3)/(sqrt(x^2+100))

I did the following work, but it seems my answer is not matching to the Wolfram|Alpha' answer.
Attachment 15355

int x^3/(sqrt( x^2 +100)) dx - Wolfram|Alpha

Don't use a trig sub, use a hyperbolic sub.

Let $\displaystyle x = 10\sinh{t}$ so that $\displaystyle dx = 10\cosh{t}\,dt$.

Then the integral becomes

$\displaystyle \int{\frac{x^3}{\sqrt{x^2 + 100}}\,dx}$

$\displaystyle = \int{\frac{1000\sinh^3{t}}{\sqrt{(10\sinh{t})^2 + 100}}\,10\cosh{t}\,dt}$

$\displaystyle = \int{\frac{1000\sinh^3{t}}{10\cosh{t}}\,10\cosh{t} \,dt}$

$\displaystyle = \int{1000\sinh^3{t}\,dt}$

$\displaystyle = \int{1000\sinh{t}\sinh^2{t}\,dt}$

$\displaystyle = \int{1000\sinh{t}(\cosh^2{t} - 1)\,dt}$

$\displaystyle = 1000\int{\sinh{t}(\cosh^2{t} - 1)\,dt}$

Let $\displaystyle u = \cosh{t}$ so that $\displaystyle du = \sinh{t}\,dt$

The integral becomes

$\displaystyle 1000\int{u^2 - 1\,du}$

$\displaystyle = 1000\left(\frac{1}{3}u^3 - u\right) + C$

Now convert it back to a function of $\displaystyle x$.