High School Math : Finding Indefinite Integrals

Study concepts, example questions & explanations for High School Math

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Example Questions

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Example Question #1 : Finding Indefinite Integrals

Possible Answers:

Correct answer:

Explanation:

The integral of  is .  The constant 3 is simply multiplied by the integral.  

Example Question #2 : Finding Indefinite Integrals

Possible Answers:

Correct answer:

Explanation:

To integrate , we need to get the two equations in terms of each other. We are going to use "u-substitution" to create a new variable, , which will equal .

Now, if , then 

 

Multiply both sides by  to get the more familiar: 

Note that our , and our original equation was asking for a positive .

That means if we want  in terms of , it looks like this:

Bring the negative sign to the outside:

.

We can use the power rule to find the integral of :

Since we said that , we can plug that back into the equation to get our answer:

Example Question #3 : Finding Indefinite Integrals

Evaluate the integral below:

 

Possible Answers:

1

Correct answer:

Explanation:

In this case we have a rational function as , where

and

 

can be written as a product of linear factors:

 

 

It is assumed that A and B are certain constants to be evaluated. Denominators can be cleared by multiplying both sides by (x - 4)(x + 4). So we get:

 

First we substitute x = -4 into the produced equation:

Then we substitute x = 4 into the equation:

Thus:

 

Hence:

 

 

Example Question #4 : Finding Indefinite Integrals

What is the indefinite integral of ?

Possible Answers:

Correct answer:

Explanation:

To solve for the indefinite integral, we can use the reverse power rule. We raise the power of the exponents by one and divide by that new exponent. For this problem, that would look like:

Remember, when taking an integral, definite or indefinite, we always add , as there could be a constant involved.

Example Question #5 : Finding Indefinite Integrals

What is the indefinite integral of ?

Possible Answers:

Correct answer:

Explanation:

To solve for the indefinite integral, we can use the reverse power rule. We raise the power of the exponents by one and divide by that new exponent. For this problem, that would look like:

Remember, when taking an integral, definite or indefinite, we always add , as there could be a constant involved.

Example Question #6 : Finding Indefinite Integrals

What is the indefinite integral of ?

Possible Answers:

Correct answer:

Explanation:

To solve for the indefinite integral, we can use the reverse power rule. We raise the power of the exponents by one and divide by that new exponent.

We're going to treat  as , as anything to the zero power is one.

For this problem, that would look like:

Remember, when taking an integral, definite or indefinite, we always add , as there could be a constant involved.

Example Question #7 : Finding Indefinite Integrals

What is the anti-derivative of ?

Possible Answers:

Correct answer:

Explanation:

To find the indefinite integral of our expression, we can use the reverse power rule.

To use the reverse power rule, we raise the exponent of the  by one and then divide by that new exponent.

First we need to realize that . From there we can solve:

When taking an integral, be sure to include a  at the end of everything.  stands for "constant". Since taking the derivative of a constant whole number will always equal , we include the  to anticipate the possiblity of the equation actually being  or  instead of just  .

Example Question #8 : Finding Indefinite Integrals

What is the indefinite integral of ?

Possible Answers:

Correct answer:

Explanation:

To find the indefinite integral of our equation, we can use the reverse power rule.

To use the reverse power rule, we raise the exponent of the  by one and then divide by that new exponent.

Remember that, when taking the integral, we treat constants as that number times  since anything to the zero power is . For example, treat  as .

When taking an integral, be sure to include a  at the end of everything.  stands for "constant". Since taking the derivative of a constant whole number will always equal , we include the  to anticipate the possiblity of the equation actually being  or  instead of just  .

Example Question #9 : Finding Indefinite Integrals

What is the indefinite integral of ?

Possible Answers:

Correct answer:

Explanation:

To find the indefinite integral of our equation, we can use the reverse power rule.

To use the reverse power rule, we raise the exponent of the  by one and then divide by that new exponent.

When taking an integral, be sure to include a  stands for "constant". Since taking the derivative of a constant whole number will always equal , we include the  to anticipate the possiblity of the equation actually being  or  instead of just  .

Example Question #10 : Finding Indefinite Integrals

What is the indefinite integral of ?

Possible Answers:

Undefined

Correct answer:

Explanation:

To find the indefinite integral of our equation, we can use the reverse power rule.

To use the reverse power rule, we raise the exponent of the  by one and then divide by that new exponent.

Remember that, when taking the integral, we treat constants as that number times , since anything to the zero power is . Treat  as .

When taking an integral, be sure to include a  stands for "constant". Since taking the derivative of a constant whole number will always equal , we include the  to anticipate the possiblity of the equation actually being  or  instead of just  .

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