# Calculus 2 : Integrals

## Example Questions

### Example Question #981 : Integrals

Evaluate the following integral:

Explanation:

To evaluate the integral, we can first use the fact that cosine and secant are inverses of each other, so they cancel:

Now, we must make the following substitution:

Rewriting the integral in terms of u and integrating, we get

We used the following rule to integrate:

Finally, replace u with our original x term:

### Example Question #982 : Integrals

Explanation:

To integrate this problem, you have to use "u" substitution. Assign . Then, find du, which is 2x. That works out since we can then replace the other x in the original problem. We will have to offset the 2 though: . Now plug in all the parts: . Now, integrate as normal, remembering to raise the exponent by 1 and then also putting that result on the bottom: . Simplify, add a C because it is an indefinite integral, and substitute your original expression back in: .

### Example Question #41 : Solving Integrals By Substitution

Explanation:

To integrate this problem, use "u" substitution. Assign , . Substitute everything in so you can integrate: . Recall that when there is a single variable on the denominator, the integral is ln of that term. Therefore, after integrating, you get . Sub back in your original expression and add C because it is an indefinite integral: .

### Example Question #983 : Integrals

Evaluate the following integral using the substitution method:

Explanation:

Make the substitution:

### Example Question #51 : Solving Integrals By Substitution

Solve: .

Explanation:

Substitute :

which is equal to

.

Replace u with 10x:

.

Explanation:

Substitute :

.

Replace :

.

Solve .

Explanation:

Substitute :

.

Replace :

.

Solve .

Explanation:

Substitue .

.

Replace :

.

Find .

Explanation:

Substitute :

.

Replace :

Solve .