# High School Math : Finding Definite Integrals

## Example Questions

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Evaluate:

Explanation:

### Example Question #2 : Integrals

Find

Explanation:

This is most easily solved by recognizing that .

### Example Question #3 : Finding Definite Integrals

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 : Integrals

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 : Integrals

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 : Integrals

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 : Integrals

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 : Integrals

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 #53 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

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 #55 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

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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