AP Calculus AB : Antiderivatives by substitution of variables

Example Questions

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Example Question #51 : Antiderivatives By Substitution Of Variables

Calculate the following integral:

Explanation:

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:

Example Question #52 : Antiderivatives By Substitution Of Variables

Calculate the following integral:

Explanation:

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:

Example Question #53 : Antiderivatives By Substitution Of Variables

Calculate the following definite integral:

Explanation:

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:

Example Question #54 : Antiderivatives By Substitution Of Variables

Calculate the following integral:

Explanation:

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:

Example Question #55 : Antiderivatives By Substitution Of Variables

Calculate the following integral:

Explanation:

Use the addition and subtraction properties of integrals to break the integral into 3 separate integrals:Evaluate the second integral:

Re-write the third integral as follows:

Evaluate the third integral:

Make the following substitution to evaluate the first integral:

Apply the substitution to the first integral:

Evaluate the integral:

Substitute the value for u back into the equation:

Combine the answers of the three integrals into one equation:

Solution:

Example Question #56 : Antiderivatives By Substitution Of Variables

Calculate the following integral:

Explanation:

Make the following substitution:

Apply the substitution to the integrand:

Evaluate the integral:

Re-substitute the value of u:

Solution:

Example Question #57 : Antiderivatives By Substitution Of Variables

Evaluate the integral

Not integrable

Explanation:

There are a lot of pieces inside this integral. There are trigonometry functions, exponential functions, and a rational arrangement. Lots of possibilities.

This is where u-substitution is best. Try making u represent various parts and see if du gets all the other parts. After doing this enough times, you will see that we should make u be the exponent of the e.

Let's write the inner fraction as an x with a negative exponent

Now we differentiate and see what we get for du. This will require the chain rule. The outer structure is the , which requires trig functions integration rules, and the chained inner structure is a power rule arrangement

Recall these derivatives:

Applying these we get

Simplifying gives us

This perfectly matches all remaining parts of our integral. Lets rewrite everything using u and du

This matches the basic integral form,

Thus, when we integrate we get

Rewriting back in terms of x, we get

Example Question #58 : Antiderivatives By Substitution Of Variables

Evaluate the integral

Explanation:

There are two ways to evaluate this integral:

(1) Rewrite in terms of sine and cosine and evaluate

(2) Break up the into and

In this explanation, we will do option (2)

Breaking up the , we get the following integral

Notice that the part is the derivative of . Lets use this fact and make the following u-substitution

Differentiating to find du, we use the chain rule and power rule:

This du accounts for all the other parts of the integral. Rewriting in terms of u, we get

This follows the basic integral form, .

Applying this rule, we get

Now we "un-substitute" back to x terms, using our original substitution, .

This gives us

Example Question #59 : Antiderivatives By Substitution Of Variables

Evaluate the integral

Explanation:

This integral can be evaluated using u-substitution. There aren't any groups raised to an exponent so we won't try the power rule for integration. Instead we notice that the integral is still a giant fraction. We will try to match this basic integral form:

Let's make the denominator our .

Differentiate to get .

This may not look like the numerator, but if we factor out an x, it will.

Now we have all the pieces matched perfectly. Let's make to substitution.

Applying the basic integral form from before, we get

Putting everything back in terms of x, we get

.