Question

How do you integrate `int x^2 /sqrt( 16+x^4 )dx` using trigonometric substitution?

Answer 1

This cannot be integrated using elementary functions.

Explanation:

Use the substitution `x^2=4tantheta`. This implies that `2xdx=4sec^2thetad theta`. Also keep in mind that `x=2sqrttantheta`.

We have:

`intx^2/sqrt(16+x^4)dx=1/2int(x(2xdx))/sqrt(16+x^4)`

`=1/2int(2sqrttantheta(4sec^2thetad theta))/sqrt(16+16tan^2theta)=int(sqrttantheta(sec^2theta)d theta)/sqrt(1+tan^2theta)`

Note that `1+tan^2theta=sec^2theta`, so `sectheta=sqrt(1+tan^2theta)`:

`=int(sqrttantheta(sec^2theta)d theta)/sectheta=intsqrttanthetasecthetad theta`

The more we continue, we see that this cannot be integrated using elementary functions.