High School Math : Integrals

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #31 : Calculus Ii — 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:

Example Question #32 : Calculus Ii — 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:

Example Question #11 : Finding Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can't use the power rule. Instead we must use u-substituion.  If 

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 #31 : Calculus Ii — Integrals

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

Since our , we can use the reverse power rule to find that the antiderivative is:

 

Remember to include a  for any integral or antiderivative 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 #35 : Calculus Ii — Integrals

If n is a positive integer, find .

 

Possible Answers:

0

 

Correct answer:

Explanation:

We can find the integral using integration by parts, which is written as follows:

 

 

Let and . We can get the box below:

 

Now we can write:

 

 

 

 

Example Question #41 : Calculus Ii — Integrals

?

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus! If , then .

Since we're given , we need to find the indefinite integral of the equation to get .

To solve for the indefinite integral, we can use the reverse power rule. We raise the power of the exponents by one and divide by that new exponent. For this problem, that would look like:

Remember, when taking an integral, definite or indefinite, we always add , as there could be a constant involved.

Now we can plug that back in:

Notice that the 's cancel out.

Plug in our given numbers.

Example Question #41 : Calculus Ii — Integrals

?

Possible Answers:

Correct answer:

Explanation:

Remember the fundamental theorem of calculus! If , then .

Since we're given , we need to find the indefinite integral of the equation to get .

To solve for the indefinite integral, we can use the reverse power rule. We raise the power of the exponents by one and divide by that new exponent.

We're going to treat  as , as anything to the zero power is one.

For this problem, that would look like:

Remember, when taking an integral, definite or indefinite, we always add , as there could be a constant involved.

Plug that back into the FTOC:

Notice that the 's cancel out.

Plug in our given values from the problem.

Example Question #11 : Finding Integrals

Possible Answers:

Correct answer:

Explanation:

Use the Fundamental Theorem of Calculus: If , then .

Therefore, we need to find the indefinite integral of our equation first.

To do that, we can use the anti-power rule or reverse power rule. We raise the exponent on the variables by one and divide by the new exponent.

For this problem, we'll treat  as  since anything to the zero power is one.

Since the derivative of any constant is , when we take the indefinite integral, we add a  to compensate for any constant that might be there.

From here we can simplify.

That means that .

Notice that the 's cancel out.

From here, plug in our numbers.

Example Question #71 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Possible Answers:

Correct answer:

Explanation:

Use the Fundamental Theorem of Calculus: If , then .

Therefore, we need to find the indefinite integral of our equation first.

To do that, we can use the anti-power rule or reverse power rule. We raise the exponent on the variables by one and divide by the new exponent.

For this problem, we'll treat  as  since anything to the zero power is one.

Since the derivative of any constant is , when we take the indefinite integral, we add a  to compensate for any constant that might be there.

From here we can simplify.

According to FTOC:

Notice that the 's cancel out. 

Plug in our given information and solve.

Example Question #41 : Calculus Ii — Integrals

Possible Answers:

Undefined

Correct answer:

Explanation:

Use the Fundamental Theorem of Calculus: If , then .

Therefore, we need to find the indefinite integral of our equation first.

To do that, we can use the anti-power rule or reverse power rule. We raise the exponent on the variables by one and divide by the new exponent.

For this problem, we'll treat  as  since anything to the zero power is one.

Since the derivative of any constant is , when we take the indefinite integral, we add a  to compensate for any constant that might be there.

From here we can simplify.

According to FTOC: 

Notice that the 's cancel out. 

Plug in our given numbers and solve.

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