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Example Questions
Example Question #91 :Solving Integrals By Substitution
Evaluate the integral with a substitution,
Let
We can now convert this back to a function ofby substituting,
Example Question #92 :Solving Integrals By Substitution
Calculate the following integral:
Add 2 and subtract 2 from the numerator of the integrand:.
Simplify and apply the difference rule:
Solve the first integral:.
Make the following substitution to solve the second integral:
Apply the substitution to the integral:
Solve the integral:
Combine the answers to the two integrals:.
Solution:
Example Question #93 :Solving Integrals By Substitution
Evaluate the Integral:
We use substitution to solve the problem:
Letand
Therefore:
Example Question #94 :Solving Integrals By Substitution
Evaluate
Here we use substitution to solve for the integrand. Let u=sin(x) therefore du= cos(x)dx. Plug your values back in:
Example Question #95 :Solving Integrals By Substitution
To integrate this expression, you have to use u substitution. First, assign your u expression:
Now, plug everything back in so you can integrate:
Now integrate:
From here substitute the original variable back into the expression.
Evaluate at 2 and then 1.
Subtract the results:
Example Question #96 :Solving Integrals By Substitution
Calculate the following integral:
Factor outfrom the integrand, and simplify:
Make the following substitution:
Plug the substitution into the integrand:
.
Use the Pythagorean identity to make the following substitution, and simplify:
Apply the following identity to the integrand:
:
.
Separate the integral into two separate integrals:
.
Solve the first integral:
.
Make the following substitution for the second integral:
.
Apply the substitution, and solve the integral:
.
Combine answers for both integrals:
Solve for:
Plug values forback into solution to integral:
Recall that,
and from above,
Therefore,
.
Example Question #97 :Solving Integrals By Substitution
Evaluate the integral:
You must use u substitution to solve this problem.
In this case
and
.
So the integral simplifies to
Example Question #98 :Solving Integrals By Substitution
Calculate the following integral:
Useas a substitute, then:
;
Now, rewrite the boundaries of integration in terms of t:
Rewrite the integral in terms of t:
Example Question #99 :Solving Integrals By Substitution
Evaluate the following integral:
We must use substitution to solve this integrand.
Let
and
don't forget to divide buy -1 to isolate the right side
putting all the values back in and pulling the negative sign out of the integral we get:
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