To integrate, the following substitution was made:

Now, we rewrite the integral in terms of u and integrate:

The following rule was used for integration:

Finally, rewrite the final answer in terms of our original x term:

First, simplifying the given's gives us

And

Our goal is to get the given expression into terms of these two integrals. Our first step will be to try and get a from our expression.

First note,

And for the third term,

Putting these facts together, we can rewrite the original expression as

Rearranging,

The three terms in parentheses can all be brought together, as the top limit of the previous integral matches the bottom limit of the next integral. Thus, we now have

Substituting in our given's, this simplifies to

To answer this, we will need to FOIL our function first.

Now can find the antiderivatives of each of these three summands using the power rule.

(Don't forget )!

is the derivative of . Thus, the antiderivative of is .

Integrate

1) Apply the sum rule for integration,

2) Integrate each individual term and include a constant of integration,

Further Discussion

Since indefinite integration is essentially a reverse process of differentiation, check your result by computing its' derivative.

This is the same function we integrated, which confirms our result. Also, because the derivative of a constant is always zero, we must include "C" in our result since any constant added to any function will produce the same derivative.