This is a Fundamental Theorem of Calculus problem.  Since a derivative and anti-derivative cancel each other out, we simply have to plug the limits into our function (with the outside variable).  Then, we multiply each by the derivative of the bound:

Using the Fundamental Theorem of Calculus, the derivative of an anti-derivative simply gives us the function with the limits plugged in multiplied by the derivative of the respective bounds:

In the last step, we made use of the following trigonometric identity:

Evaluate the following indefinite integral:

Recall that we can split subtraction and addition within integrals into separate integrals. This means that we can look at our problem in two steps.

Recall that we can integrate any exponential term by adding 1 to the exponent and dividing by the new exponent.

So,

Next, recall that the integral of sine is negative cosine. However, we already have a negative sine, so we should get positive cosine.

Now, we can combine our two halves to get our final answer. 

Notice that we only have one "c" because c is just a constant, not a variable.

Evaluate the given definite integral:

Let's look at this integral as three separate parts, then we'll combine those parts to evaluate the answer.

First, the integral of  is just 

Second, The integral of any exponential term can be found by increasing the exponent by 1 and then dividing by the new number.

Third, the integral of cosine is simply positive sine.

Put all this together to get:

Now, we need to evaluate our answer by finding the difference between the limits of integration.

First, let's see what we get when we plug in 0

Next, let's use 5

Next, find the difference of F(5)-F(0)

So our answer is: -2388.89.

Evaluate the following definite integral.

Lets begin by recalling that taking an integral is essentially the same as reversing the differentiation process.

So, when we integrate sine, we get negative cosine, and when we integrate cosine, we get sine. The coefficients will stay the same.

Now, we have integrated, but we still need to evaluate our integral. To do so, we need to find the difference between  and .

Now, find the difference between our two values:

So, our answer must be 7.

Derivatives and anti derivatives annihilate each other.  Therefore, the derivative of an anti derivative is simply the function in the integral with the limits substituted in multiplied by the derivative of each limit.  Also, be careful that the units will match the outermost units (which comes from the derivative).

We will use the Fundamental Theorem of Calculus

First we find the anti derivative using the rule, then we apply the FTC

Now we just need to simplify,

Here we use the Fundamental Theorem of Calculus

and the rule

First we find the anti-derivative using the rule above (no constant is needed because we are dealing with definite integrals)

Then we use the FTC by plunging in 3 and subtracting from plugging in 0

We will use the Fundamental Theorem of Calculus 

and the rule

First we find the anti derivative

And then we evaluate, (upper minus lower)

(Remembering your logarithm rules)

We will use the Fundamental Theorem of Calculus

First we find the anti derivative

Then we evaluate it (upper minus lower)