All Calculus 2 Resources
Example Questions
Example Question #651 : Finding Integrals
Solve .
For
,
first substitute
.
Replace u with :
.
Example Question #651 : Finding Integrals
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 #652 : Finding Integrals
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 #653 : Finding Integrals
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 #654 : Finding Integrals
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 .
Remember to add a C because it is an indefinite integral. Your final answer is:
.
Example Question #1005 : Integrals
Solve the indefinite integral
Hint: use u-substitution
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
and replacing yields the answer
Example Question #71 : Solving Integrals By Substitution
Solve:
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 #1006 : Integrals
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 #1007 : Integrals
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 #1008 : Integrals
Calculate the following integral:
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:
.
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