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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 6

Lesson 8: Finding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule

# Reverse power rule review

Review your knowledge of the reverse power rule for integrals and solve problems with it.

## What is the reverse power rule?

The reverse power rule tells us how to integrate expressions of the form x, start superscript, n, end superscript where n, does not equal, minus, 1:
integral, x, start superscript, n, end superscript, d, x, equals, start fraction, x, start superscript, n, plus, 1, end superscript, divided by, n, plus, 1, end fraction, plus, C
Basically, you increase the power by one and then divide by the power plus, 1.
Remember that this rule doesn't apply for n, equals, minus, 1.
Instead of memorizing the reverse power rule, it's useful to remember that it can be quickly derived from the power rule for derivatives.

## Integrating polynomials

We can use the reverse power rule to integrate any polynomial. Consider, for example, the integration of the monomial 3, x, start superscript, 7, end superscript:
\begin{aligned} \displaystyle\int 3x^7\,dx&=3\left(\dfrac{x^{7+1}}{7+1}\right)+C \\\\ &=3\left(\dfrac{x^8}{8}\right)+C \\\\ &=\dfrac{3}{8}x^8+C \end{aligned}
Problem 1
• Current
integral, 14, t, d, t, equals, question mark

Want to try more problems like this? Check out these exercises:

## Integrating negative powers

The reverse power rule allows us to integrate any negative power other than minus, 1. Consider, for example, the integration of start fraction, 1, divided by, x, squared, end fraction:
\begin{aligned} \displaystyle\int \dfrac{1}{x^2}\,dx&=\displaystyle\int x^{-2}\,dx \\\\ &=\dfrac{x^{-2+1}}{-2+1}+C \\\\ &=\dfrac{x^{-1}}{-1}+C \\\\ &=-\dfrac{1}{x}+C \end{aligned}
Problem 1
• Current
integral, 8, t, start superscript, minus, 3, end superscript, d, t, equals

Want to try more problems like this? Check out these exercises:

## Integrating fractional powers and radicals

The reverse power rule also allows us to integrate expressions where x is raised to a fractional power, or radicals. Consider, for example, the integration of square root of, x, end square root:
\begin{aligned} \displaystyle\int \sqrt x\,dx&=\displaystyle\int x^{^{\large\frac{1}{2}}}\,dx \\\\ &=\dfrac{x^{^{\large\frac{1}{2}\normalsize+1}}}{\dfrac{1}{2}+1}+C \\\\ &=\dfrac{x^{^{\large\frac{3}{2}}}}{\frac{3}{2}}+C \\\\ &=\dfrac{2\sqrt{x^3}}{3}+C \end{aligned}
Problem 1
• Current
integral, 4, t, start superscript, start fraction, 1, divided by, 3, end fraction, end superscript, d, t, equals, question mark

Want to try more problems like this? Check out these exercises:

## Want to join the conversation?

• What would you do if n=-1? What would the integral be of x^-1?
• ln(x)

If you have mastered differential calc at KA, then you most likely have come across the derivative of ln(x) = 1/x.
• It's so easy.
Just like differential calculus, integral calculus has its own rules.
• Until when is integration going to be fun like this?
• It's fun until you enjoy it. Integrals will get lengthier and will require more methods and thinking, but if you learn to enjoy a subject, no matter how hard concepts get, you'll still have fun doing problems.

Frankly though, integrals in Calc II can get pretty nasty as here, they're testing you on how smart you are when it comes to figuring out a way to integrate. So, they can give you the wackiest integrals on the planet. If/when you reach Calc III, you'll learn about double and triple integrals (Yeah. They exist lol!). Here, you're not tested on how well you can integrate. So, your integrand will be fairly simple. You'll be tested on how well you can visualize and define the region (which is easy in single variable Calculus as there is only one axis to take care of), which is really the hardest part about multiple integrals.
• How do we go from 1/3 x^3 to 4sqrtx^8?
• 4sqrtx^8 is rewritten as x^2, because (x^2)^4 = x^8
Therefore, the antiderivative of x^2 is:
x^(2+1) / (2+1) + C
x^3 / (3) + C
1/3 x^3 + C
• how do you integrate sin^2 x?
• What is integral of √ax+b dx
(1 vote)
• It is ambiguous.....are both ax under the radical or just a?....Let's solve the first case which is the most laborious case....

2*5^(1/2)*x^(3/2)/3 + bx I hope it helps!
• how do you do this
• how do you integrate>> ((4with under root) x^8)