Calculus/Indefinite integral/Solutions

1. Evaluate
We need to find a function, , such that

We know that

So we need to find a constant, , such that

Solving for , we get

So

Check your answer by taking the derivative of the function you've found and checking that it matches the integrand:

We need to find a function, , such that

We know that

So we need to find a constant, , such that

Solving for , we get

So

Check your answer by taking the derivative of the function you've found and checking that it matches the integrand:

2. Find the general antiderivative of the function .
We know that


We need to find a constant, , such that

Solving for , we get

So the general antiderivative will be

Check your answer by taking the derivative of the antiderivative you've found and checking that you get back the function you started with:

We know that


We need to find a constant, , such that

Solving for , we get

So the general antiderivative will be

Check your answer by taking the derivative of the antiderivative you've found and checking that you get back the function you started with:

3. Evaluate
4. Evaluate
5. Evaluate by making the substitution
Since , and
Since , and
6. Evaluate
Let , so that
Let , so that
7. Evaluate using integration by parts with and
;
;
8. Evaluate
Let ;

Then and
To evaluate , make the substitution ; ; . Then
. So

Let ;

Then and
To evaluate , make the substitution ; ; . Then
. So