## Answers

#### Explanation:

Complete the square,

Substitute

Substitute

Simplify,

Refine,

Take out the constant,

Apply double angle formulae,

Take out the constant,

Integrate,

Substitute back

Simplify,

Refine,

Tadaa :D

#### Explanation:

What is

Note that the domain of the function being integrated is where the inner quadratic is positive, i.e.

This expression can be integrated using substitutions. Though a possible pathway for integration doesn't immediately present itself, if we compete the square, then a trigonometric substitution can be carried out:

Which, we notice, is in the classic trigonometric substitution form, i.e. the square of a number minus the square of a linear

First, to get rid of the linear, we let

Now for the second substitution, let

Of course, the

Now we can use a double angle formula to make integrating

So the integral becomes:

Now,

Hence,

And,