# Calculus 1 : How to find integral expressions

## Example Questions

### Example Question #2191 : Calculus

Evaluate the integral:

Explanation:

To integrate, we must make the following subsitution:

Now, rewrite the integral, and integrate:

The following rule was used for the integration:

Finally, replace u with our original term:

Note that the absolute value sign went away, because the square root is always positive.

### Example Question #2192 : Calculus

Evaluate the following integral:

Explanation:

To integrate, we must split the integral in two:

The first integral is:

and is identical to the rule used to integrate.

The second integral is performed using the following substitution:

Now, rewrite the integral and integrate:

We used the following rule to integrate:

Finally, replace u with the original term:

### Example Question #2193 : Calculus

Evaluate the following integral:

Explanation:

The integral is equal to

and was found using the following rules:

.

### Example Question #2194 : Calculus

Evaluate the following integral:

Explanation:

The integral is equal to

and we used the following rules for integration:

### Example Question #2194 : Calculus

Evaluate the integral:

Explanation:

The integral was performed using the following rules:

Applying the above rules we get the following.

### Example Question #2195 : Calculus

Evaluate the following integral:

Explanation:

To integrate, we must make the following substitution:

Now, rewrite the integral and integrate:

The following rule was used to integrate:

Finally, replace u wiht our original term:

### Example Question #2196 : Calculus

Explanation:

The integral can be split into two seperate ones:

The first integral is simple:

and is identical to the rule used.

The second integral can be performed using the following substitution:

Rewrite and integrate:

The following rule was used to integrate:

Finally, replace u with the original term and add the integrals together:

### Example Question #2197 : Calculus

Given that the piecewise function  for  and  for , find .

Explanation:

Since we have a piecewise function we will need to set up an integral with two parts. One will  from  to zero and the other,  from zero to two.

Setting up the integral and plugging in the bounds looks like,

.

### Example Question #119 : Integral Expressions

Find the general anti-derivative of .

Explanation:

We have a product of two easy functions, so to get our answer we need to use integration by parts. The general formula for this is:

.

Choose  and . Then  and

So integration by parts tells us that our antiderivative equals

Now we must apply integration by parts again, because we still have an integral of a product of two functions. Choose  and  again, so  and .

Then we have

But of course,  is its own antiderivative, because we can pull out constants and  is its own anti-derivative.

So combining these two results we get our final answer:

### Example Question #2198 : Calculus

Write the integral expression for position  given acceleration