High School Math : Finding Definite Integrals

Study concepts, example questions & explanations for High School Math

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Example Questions

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Example Question #1 : Finding Definite Integrals

Evaluate:

Possible Answers:

Correct answer:

Explanation:

 

Example Question #2 : Finding Integrals

Find  

Possible Answers:

Correct answer:

Explanation:

This is most easily solved by recognizing that .  

Example Question #2 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can't use the power rule. Instead we end up with: 

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

Example Question #3 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can use the reverse power rule to find the indefinite integral or anti-derivative of our function:

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

Example Question #4 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

As it turns out, since our , the power rule really doesn't help us.  has a special anti derivative: .

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

Example Question #5 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

As it turns out, since our , the power rule really doesn't help us.  is the only function that is it's OWN anti-derivative. That means we're still going to be working with .

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

Because  is so small in comparison to the value we got for , our answer will end up being 

Example Question #6 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can use the power rule for all of the terms involved to find our anti-derivative:

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

Example Question #7 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can't use the power rule. We have to break up the quotient into separate parts:

 

.

The integral of 1 should be no problem, but the other half is a bit more tricky:

 is really the same as . Since ,  .

Therefore:

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

Example Question #8 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can use the power rule, if we turn it into an exponent: 

This means that:

 

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

Example Question #9 : Finding Definite Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can't use the power rule, as it has a special antiderivative:

Remember to include the  for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

Notice that the c's cancel out. Plug in the given values for a and b and solve:

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