The fundamental theorem of calculus states that if , then .

First, we need to find the indefinite integral of our given equation. Just like with the derivatives, the indefinite integrals or anti-derivatives of trig functions must be memorized.

To solve for the definite integral, use the fundamental theorem of calculus. If , then .

First we need to find the indefinite integral.

We can solve this problem using the Fundamental Theorem of Calculus:Iif , then .

To use that equation for this problem, we need to find the indefinite integral of our given equation.

To find the indefinite integral of , we can use the reverse power rule. To do this, we raise our exponent by one and then divide the variable by that new exponent.

Don't forget to include a  to cover any constant!

Now we can plug that into the FTOC:

Notice that the 's cancel out.

Plug in our given values:

To solve this problem we can use the Fundamental Theorem of Calculus: If , then .

First we need to find the indefinite integral.

Since we're adding terms, we take the integral or anti-derivative of each part separately. For , we can use the reverse power rule, which states that we raise the exponent of the variable by one and then divide by that new exponent. For sine, we use our trigonometric integral rules.

Remember, .

Don't forget to include a  to account for any constants!

Plug that into the FTOC:

Notice that the 's cancel out.

Plug in our given values:

To find the definite integral, we can use the Fundamental Theorem of Calculus. This states that if , then .

To use the FToC, we need to find our indefinite integral of our given equation.

To find the indefinite integral, or anti-derivative, we can use the reverse power rule. We raise the exponent of each variable by one and divide by that new exponent.

Don't forget to include a  to cover any constant!

Simplify.

We can now plug that into FToC!

Notice that the 's cancel out.

Plug in our given values.

Remember the Rundamental Theorem of Calculus: If , then .

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 into the problem.

Notice that the 's cancel out. Plug in the values given in the problem:

The integral of  is .  The constant 3 is simply multiplied by the integral.  

To integrate , we need to get the two equations in terms of each other. We are going to use "u-substitution" to create a new variable, , which will equal .

Now, if , then 

Multiply both sides by  to get the more familiar: 

Note that our , and our original equation was asking for a positive .

That means if we want  in terms of , it looks like this:

Bring the negative sign to the outside:

.

We can use the power rule to find the integral of :

Since we said that , we can plug that back into the equation to get our answer:

In this case we have a rational function as , where

and

can be written as a product of linear factors:

It is assumed that A and B are certain constants to be evaluated. Denominators can be cleared by multiplying both sides by (x - 4)(x + 4). So we get:

First we substitute x = -4 into the produced equation:

Then we substitute x = 4 into the equation:

Thus:

Hence:

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.