Calculus/Indefinite integral/Solutions

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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:

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:

3. Evaluate

4. Evaluate

5. Evaluate by making the substitution

Since , and

6. Evaluate

Let , so that

7. Evaluate using integration by parts with and

;

8. Evaluate

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