Solve by u substitution

Make the following substitution:  

Apply the substitution the integral: 

Solve the integral: 

Re-substitute the value for u: 

To integrate, we can split the integral into the sum of two separate integrals:

We can solve them independently and add the results together.

To solve the first integral, we make the following substitution:

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

The integral was found using the identical rule.

Rewriting the integral in terms of x, we get

We solve the second integral using the following substitution:

Rewriting the integral in terms of u, we get

The integral was performed using the identical rule.

Now, rewrite the answer in terms of x:

Combining the results from each integral, we get

Note that the two constants of integration combine to make a single constant. 

To integrate, the following substitution must be made:

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

The integration was performed using the following rule:

Finally, we rewrite our answer in terms of x:

Evaluating integrals requires knowledge of the basic integral forms. In this problem, there is a group raised to an exponent, which is the .

This arrangement typically follows the basic integral form , where is a variable expression, is some constant number and is the constant of integration, which stays unknown. In this case, we can use u-substitution to match this form.

The exponent is 5, so the in the basic integral form will be5.

Since the inside of the exponent group is , set .

Now we differentiate to find . Recall that the derivative of .

Notice that the parts of our match up with the integral we are evaluating, and there are no variables that aren't accounted for. This is clearer if we write out and simplify what our substitution says.

This is exactly what we are asked to integrate.

This means we can evaluate the integral by using basic integral form directly. Plugging in our and into the right side of  , we get

This is equivalent to the answer , which is the correct answer.

The first thing to recognize about this integral is that the derivative of is .  is in the denominator and is in the numerator. This arrangement usually results in the natural log, , of the bottom. The basic integral form is as follows

This means we need to set the denominator as in our substitution.

Now we find by differentiating . Remember that the derivative of is . This derivative involves chain rule and power rule.

This is similar to the numerator. We have the variables but the is not present in the given integral. Fortunately, is a constant and can be corrected by dividing our  equation by  on both sides. Doing so yields

This adjustment means that when we plug our information into the basic integral form, we will write instead of just . Doing so, we get

The is a constant, and thus can pass to the left of the integral sign.

From the basic intergral form, the is equal to . So we replace just the integral with the equivalent answer, but we leave the .

Plugging in 's substitution, we get

The last thing we do is distribute the to the natural log and the constant, .

The constant, , is unknown, so when we multiply it by , it is still unknown, so we leave it as .This yields

, which is the correct answer.

To evaluate this integral requires the basic integral form, .

In this form, the is the exponent of . For our problem, this exponent is . Thus, we will set for our u-substitution. 

Now we need to find and verify that it contains all the other variables in our integral. Finding the derivative of  will require the Chain rule. Differentiating , we get

This matches all the variables not accounted for by the , but our has an extra factor of 4. We address this by dividing both sides by 4 to match what we have in our integral.

Now we can rewrite the original integral using variables.

Since the is a constant, we can pass it to the left of the integral sign. Then we will apply the basic integral form to the remaining integral.

When we distribute the to the inside of the group, remember the the will still be unknown, so it stays as is.

Finally, we "un-substitute" our for what it equals originally, .

This is the correct answer.

Factor out the constant: 

Re-write tanx in terms of sine and cosine: 

Make the following substitution:     

Apply the substitution to the integrand: 

Evaluate the integral: 

Re-substitute the value for u back into the equation: 

Solution: 

Re-write the integrand as follows: 

Make the following substitution:   

Apply the substitution to the integrand: 

Evaluate the integral: 

Re-substitute the value of u: 

Solution: 

Make the following substitution:   

Use the above substitution to calculate the new limits of integration:      

Re-write the definite integral with the substitution and the new limits of integration: 

Evaluate the definite integral: 

Solution: 

Knowing the integral of a sum is equal to the sum of the integrals, break the integral into two separate integrals: 

Evaluate the first integral: 

Make the following substitutions in the second integral:     

Apply the substitutions to the integral: 

Evaluate the integral: 

Combine the results of the first and second integrals: 

Solution: