# Calculus 2 : Definite Integrals

## Example Questions

### Example Question #55 : Finding Integrals

Evaluate this integral.

Explanation:

In order to evaluate this integral, first find the antiderivative of

In this case, .

The antiderivative is  .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

### Example Question #56 : Finding Integrals

Evaluate the following definite integral:

Explanation:

By a straightforward application of antiderivative rules for polynomial functions,

### Example Question #57 : Finding Integrals

Evaluate the integral .

Explanation:

This integral is tricky, because there doesn't seem to be an apparent substitution method to use. The derivative of , and  seem to only complicate the integral further. However, we do note that the integral of  is given by . Thus, we can use integration by parts here. Splitting  into  and letting  and , this gives us  and . Plugging this into the integration by parts formula, we get:

Unfortunately, this still looks more complicated than the original. However, using the trig identity  on the second integral will give us something we can work with. Doing so we obtain:

Now here, we know how to evaluate the integral of , and we also have a copy of our integral  subtracted on the right side. Therefore, we can add this integral to each side of the equation, eliminating it on the right side and leaving something we can evaluate. Doing so, we obtain:

Finally, dividing the whole equation by  and replacing the  constant with a new constant  gives us our final result:

### Example Question #58 : Finding Integrals

Evaluate.

Explanation:

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

### Example Question #59 : Finding Integrals

Explanation:

The antiderivative of  is .  Now, all we need to do is plug in our bounds, remembering it is always top - bottom bound.

In the last step, I simply rewrote  into the denominator, getting rid of the negative exponent.

### Example Question #60 : Finding Integrals

Evaluate.

Explanation:

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

### Example Question #41 : Definite Integrals

Evaluate.

Explanation:

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

### Example Question #42 : Definite Integrals

Evaluate.

Explanation:

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

### Example Question #43 : Definite Integrals

Evaluate.

Explanation:

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

Evaluate.