# Calculus 1 : How to find integral expressions

## Example Questions

### Example Question #101 : Integral Expressions

Evaluate the following indefinite integral:

Explanation:

To solve this problem, we have to remember properties of natural logs. Pull the 2 out in front of the integral and solve. Don't forget to add "+C" at the end!

### Example Question #101 : How To Find Integral Expressions

Evaluate the following indefinite integral:

Explanation:

To solve this integral, use the power rule. Applying it to this problem gives us the following for the first term:

And the following for the second term:

We can combine these terms and add our "C" to get the final answer:

### Example Question #102 : How To Find Integral Expressions

Evaluate the following integral:

Explanation:

To integrate, we must make the following subsitution:

Rearranging to get du in terms of x we get the following.

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

which was found using the following rule:

Now, replace u with our original term:

Notice that the absolute value went away, because the square root is always positive.

### Example Question #103 : How To Find Integral Expressions

Find the integral of the function .

Explanation:

To find this integral, use the method of integration by parts:

Let  such that

There is now an integral on each side; however, it is the same integral. Move the integral on the right to the left:

Now recall that we're asked to find the value of , so to find that, we need only divide each side by two:

Ta-da.

### Example Question #104 : Equations

Evaluate the indefinite integral:

Explanation:

Begin by rewriting the equation in terms of exponents:

Afterwards, add one to each exponent and divide by the resultant value for each term to do the integral; be sure to add a constant of integration:

For this particular problem we will use the Power Rule on each term.

Power Rule: .

Appying this rule we find the following function.

### Example Question #104 : How To Find Integral Expressions

Evaluate the following integral:

Explanation:

To integrate, we must perform the following substitution:

Now, rewrite the integral and integrate:

The integration was performed using the following rule:

Finally, replace u with the original term we designated at the start:

### Example Question #105 : How To Find Integral Expressions

Evaluate the following integral:

Explanation:

To integrate, we must perform the following substiution:

Now, rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace u with our original, x term:

### Example Question #106 : How To Find Integral Expressions

Evaluate the integral:

Explanation:

To perform the integral, we must use the following substitution:

Now, rewrite the integral and integrate:

The integration was performed using the following rule:

Now, replace u with the original, x containing term:

### Example Question #107 : How To Find Integral Expressions

Evaluate the integral:

Explanation:

To integrate, we must perform the following subsitution:

Now, rewrite the integral and integrate:

We used the following rule to integrate:

Finally, replace u with our original term:

### Example Question #108 : How To Find Integral Expressions

Evaluate the integral: