## Tuesday, November 4, 2014

### Advanced Math Solutions – Integral Calculator, trigonometric substitution

In the previous posts we covered substitution, but standard substitution is not always enough. Integrals involving radicals for instance, we want to get rid of the square root.
Here’s how:

• For \sqrt{a^2-bx^2}, let x=\frac{\sqrt{a}}{\sqrt{b}}\sin \left(u\right) and use the identity 1-\sin^2(u)=\cos^2(u)
• For \sqrt{a^2+bx^2}, let x=\frac{\sqrt{a}}{\sqrt{b}}\tan \left(u\right) and use the identity 1+\tan^2(u)=\sec^2(u)
• For \sqrt{bx^2-a^2}, let x=\frac{\sqrt{a}}{\sqrt{b}}\sec \left(u\right) and use the identity \sec^2(u)-1=\tan^2(u)

From here simply cancel, integrate and substitute back

We covered almost all there is to know about substitution.  In the next post we will cover inverse trigonometric functions.

Cheers,
Michal