# Calculus 2 : Definite Integrals

## Example Questions

### Example Question #181 : Definite Integrals

Solve the definite integral by using u-substitution.

Explanation:

So first things first, we identify what our u should be. If we look at this for chain rule our inside function would be the  in the . Therefore we use this as our u.

Next we derive.

Solve for dx.

Substitute it back in.

Simplify. If all the xs don't cross out, we have done something wrong.

Integrate.

Plug in the original.

Plug in values and substract

### Example Question #182 : Definite Integrals

Evaluate.

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 #183 : Definite Integrals

Evaluate.

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 #184 : Definite Integrals

Evaluate.

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 #185 : Definite Integrals

Evaluate.

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 #186 : Definite Integrals

Explanation:

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

Now, evaluate at 2 and then 0. Subtract the results:

### Example Question #187 : Definite Integrals

Evaluate the following:

Explanation:

Solving this problem first requires knowledge of antiderivatives and their rules as well as the properties of definite integrals.

### Example Question #188 : Definite Integrals

Solve the following definite integral.

Explanation:

### Example Question #189 : Definite Integrals

If

Not enough information.

Explanation:

If we think of the anti-derivative as computing the area under the curve, then between  and , the area under our function equals .  We also know that between  and , the area under our function equals .  Then, to find the area under the function between  and , we must simply subtract the other two areas.  Therefore:

If , what is ?