Calculus 2 : Definite Integrals

Example Questions

Example Question #91 : Definite Integrals

If and , what is ?

Explanation:

First off, you're going to want to integrate the derivative to get the original function.

When integrating, raise the exponent by 1 and put that result on the denominator.

Therefore, the integral is

.

Remember to add a C because it is an indefinite integral at this point.

To find C, plug in 1 for x and set the integral equal to 2 from your initial conditions:

.

Plug 2 in for C to get your function:

.

Example Question #92 : Definite Integrals

If and what is ?

Explanation:

Recall that the integral of acceleration is velocity.

Therefore, integrate the acceleration function first. Recall that you raise the exponent by 1 and then put that result on the denominator.

Therefore,

.

Remember to add a +C because it is an indefinite integral.

Then, plug in your initial conditions to find C:

.

.

Example Question #93 : Definite Integrals

Evaluate.

Explanation:

In this case, .

The antiderivative is  .

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

Example Question #94 : Definite Integrals

Evaluate.

Explanation:

In this case, .

The antiderivative is  .

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

Example Question #95 : Definite Integrals

Evaluate.

Explanation:

In this case, .

The antiderivative is  .

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

Example Question #96 : Definite Integrals

Evaluate.

Explanation:

In this case, .

The antiderivative is  .

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

Example Question #111 : Finding Integrals

Evaluate.

Explanation:

In this case, .

The antiderivative is  .

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

Example Question #112 : Finding Integrals

Evaluate.

Explanation:

In this case, .

The antiderivative is  .

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

Example Question #113 : Finding Integrals

Explanation:

First, integrate the expression. Remember to raise the exponent by 1 and then put that result on the denominator as well: . Evaluate at 2 and then 0. Subtract the results. .

Example Question #114 : Finding Integrals

Explanation:

First, integrate the expression. Remember to raise the exponent by 1 and then also put that result on the denominator.

Therefore, after integrating, it should look like:

.

Then, first evaluate at 4 and then 0.

Subtract the results:

.