# Calculus 3 : Integration

## Example Questions

### Example Question #21 : Integration

Calculate

Explanation:

This can be a challenging integral using standard methods.  However, it is easy if we use integration by-parts, given as

Choose

.

From the definition,

### Example Question #21 : Integration

Calculate

Explanation:

This integral is most easily found by implementing u-substitution.  Choose

, which means we can rewrite the integral in a more familiar form

### Example Question #22 : Integration

Calculate

Explanation:

This integral is most easily done by using u-substitution.  Initially rewrite our integral as

, then choose

.  Therefore

### Example Question #24 : Integration

Calculate

Explanation:

The solution will be obtained by basic definitions of integrals, and properties of the natural logarithm, namely the fact that  and .

.

### Example Question #25 : Integration

Find

Can not be determined

Explanation:

This integral can most easily be done by u-substitution.  Begin by rewriting the integral as

Choose , therefore

.

### Example Question #26 : Integration

Calculate

Explanation:

This integral can be evaluated by using u-substitution.  Choose

.  This means

.

### Example Question #23 : Integration

Integrate:

Explanation:

In order to integrate, we will need to expand the binomial.  Substitution will not work since there is no valid  term to replace  if we let .

Expand the binomial.

The integral becomes:

Integrate each term.

### Example Question #28 : Integration

Integrate:

Explanation:

Use substitution to solve this question.

Let , and by adding three on both sides, and we have .

Take the derivative of u with the respect to x.

With the three equations, we can substitute the integral in terms of u.

Rewrite the denominator as , and separate the integral into two integrals.

Integrate the terms separately.

Add the two integrals.  The constants can be combined as a single constant at the end.

We can also pull out a  as a common factor.

Substitute  back into the answer.

Simplify the terms in the bracket.

The fractional exponent of one-half is a square root of the quantity.

### Example Question #322 : Calculus 3

Evaluate:

Explanation:

Step 1: Find the integration of each term:

Step 2: Re-write the function in terms of the integrated terms:

Step 3: Find the upper limit:

Step 4: Find the Lower Limit:

Step 5: Subtract the upper limit and the lower limit:

Integrate