Calculus 2 : Integrals

Example Questions

Example Question #61 : Solving Integrals By Substitution

Explanation:

To integrate this expression, use u substitution. Assign . Now, you can substitute everything in: . Remember that when integrating a single variable on the denominator, the integral is ln of that term. After integrating, you get: . Substitute the original expression back in and add a +C because it is an indefinite integral:

Example Question #61 : Solving Integrals By Substitution

Explanation:

To integrate this expression, use u substitution. Assign . Everything can be substituted, so now rewrite: . Remember that when there is a single variable on the denominator, the integral is ln of that term: . Substitute back in your initial expression: . Now, evaluate at 3 and then 1. Subtract the results:

Example Question #66 : Solving Integrals By Substitution

Explanation:

To integrate this expression, you must use u substitution.

Assign

.

Now everything can be substituted in.

The new integration problem looks like this:

.

Remember that when there is a single variable on the denominator, the integral is natural log of that term.

After integrating, you should get

.

Then substitute back in your original expression and add a +C because it is an indefinite integral:

Example Question #63 : Solving Integrals By Substitution

Explanation:

To integrate this expression, use u substitution.

Assign

.

Since everything can be substituted, rewrite the problem:

.

The integral of is .

Therefore, after integrating, you get: . Then, substitute your original expression back in to get .

.

Example Question #71 : Solving Integrals By Substitution

Solve the indefinite integral

Hint: use u-substitution

Explanation:

We first rewrite the function

To solve the indefinite integral, we set .

Deriving then gives the equation , or . Substituting in for  and  gives the integral

Finding the anti derivative of this function we get

Example Question #72 : Solving Integrals By Substitution

Solve:

Explanation:

To integrate, we must first make a substitution:

The derivative was found using the rule

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

The integral was found using the following rule:

Finally, replace u with our original x term:

Example Question #73 : Solving Integrals By Substitution

Explanation:

First, assign u substitution in order to integrate the expression:

Now, substitute everything in so you can integrate:

Now, integrate. Remember when there is a single x on the denominator, the integral is ln of that term.

Now, substitute back in the initial expression and add a +C because it is an indefinite integral:

Example Question #71 : Solving Integrals By Substitution

Explanation:

To integrate this expression, you'll have to use u substitution. Assign your "u."

Now, substitute everything in:

Integrate:

Substitute your original expression back in and add a C because it is an indefinite integral:

Example Question #74 : Solving Integrals By Substitution

Calculate the following integral:

Explanation:

To solve this integral, we use u substitution. However, to do so, we must break our integral into two separate integrals, which looks like this:

Now that we have two separate integrals, we can make the appropriate substitutions for each one. For the first integral, we make the following substitution:     . For the second integral, we make this substitution:   . This changes our integral to: , which equals:. Plugging back in our respective values for u and v, we get:

.

Example Question #76 : Solving Integrals By Substitution

Evaluate the integral:

Explanation:

There are no apparent substitutions to rewrite the integrand with other than a trigonometric substitution. The denominator resembles  which means that . Specifically,  which means that .

With this information,

The entire denominator of the integrand, excluding the radical, can be rewritten as  simply by replacing  with

This can be simplified to . This comes from the trigonometric identity .

Now, this problem can be rewritten entirely in terms of :

The integral further simplifies to:

There is no way to evaluate this integral other than rewriting the integrand as:

This comes from the trigonometric identity:

Now, the integral can be easily evaluated by splitting the integrand:

The second integral was evaluated using the following:

The integral may seem to be evaluated. However, the original integral was in terms of . Therefore, every  must be turned back to

You know from the beginning of the problem that . This can be solved in terms of  by dividing both sides of the equation by 2 and then by taking the inverse sine of both sides, leaving you with:

The only way that the second term can be rewritten in terms of  is by using . Using the fact that and . This can be found by knowing that  where  and

The second term can now be rewritten as:

This simplifies to: