All GRE Subject Test: Math Resources
Example Questions
Example Question #1 :Substitution
集成:
This problem requires U-Substitution. Letand find.
Notice that the numerator inhas common factor of 2, 3, or 6. The numerator can be factored so that theterm can be a substitute. Factor the numerator using 3 as the common factor.
Substituteandterms, integrate, and resubstitute theterm.
Example Question #1 :Substitution
Evaluate the following integral:
To calculate this integral, we could expand that whole binomial, but it would be very time consuming and a bit of a pain. Instead, let's use u substitution:
Given this:
We can say that
Then, plug it back into our original expression
Evaluate this integral to get
Then, replace u with what we substituted it for to get our final answer. Note because this is an indefinite integral, we need a plus c in it.
Example Question #1 :Substitution
集成the following using substitution.
Using substitution, we setwhich means its derivative is.
Substitutingfor, andforwe have:
Now we just integrate:
Finally, we remove our substitutionto arrive at an expression with our original variable:
Example Question #4 :Substitution
Evaluate the following integral:
To calculate this integral, we could expand that whole binomial, but it would be very time consuming and a bit of a pain. Instead, let's use u substitution:
Given this:
We can say that
Then, plug it back into our original expression
Evaluate this integral to get
Then, replace u with what we substituted it for to get our final answer. Note because this is an indefinite integral, we need a plus c in it.
Example Question #1 :Integrals
集成the following.
Integration by parts follows the formula:
So, our substitutions will beand
which meansand
Plugging our substitutions into the formula gives us:
Since, we have:
, or
Example Question #1 :Integrals
Evaluate the following integral.
Integration by parts follows the formula:
In this problem we haveso we'll assign our substitutions:
and
which meansand
Including our substitutions into the formula gives us:
We can pull out the fraction from the integral in the second part:
Completing the integration gives us:
Example Question #1 :Integrals
Evaluate the following integral.
Integration by parts follows the formula:
Our substitutions will beand
which meansand.
Plugging our substitutions into the formula gives us:
Look at the integral: we can pull out theand simplify the remainingas
.
We now solve the integral:, so:
Example Question #1 :Integration By Parts
Evaluate the following integral.
Integration by parts follows the formula:
.
Our substitutions areand
which meansand.
堵在我们替换公式投入es us
We can pulloutside of the integral.
Since,we have
Example Question #1 :Integrals
集成the following.
We can integrate the function by using substitution whereso.
Just focus on integratingsinenow:
The last step is to reinsert the substitution:
Example Question #41 :Derivatives & Integrals
集成the following.
We can integrate using substitution:
andso
Now we can just focus on integratingcosine:
Once the integration is complete, we can reinsert our substitution:
Certified Tutor
Certified Tutor