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Calculus 2
Unit 1: Lesson 10
Reverse power rule- Reverse power rule
- Reverse power rule
- Reverse power rule: negative and fractional powers
- Indefinite integrals: sums & multiples
- Reverse power rule: sums & multiples
- Rewriting before integrating
- Reverse power rule: rewriting before integrating
- Rewriting before integrating: challenge problem
- Reverse power rule review
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Reverse power rule review
AP.CALC:
FUN‑6 (EU)
, FUN‑6.C (LO)
, FUN‑6.C.1 (EK)
, FUN‑6.C.2 (EK)
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:
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.
Want to learn more about the reverse power rule? Check out this video.
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:
Remember you can always check your integration by differentiating your result!
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:
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:
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?(2 votes)
actually, ln|x| to include negative x values as well.(2 votes)
how do you integrate sin^2 x?(3 votes)
I'm wondering how to work the second part of fractional powers and radicals....specifically the 4 sqrtx^8...how did it become x^2? and also the first part, did it become x^(5/2) because I raised the function to (1/2) and then just multiply straight across on the exponents? And if I may ask one more question, On problem #3 integrating negative powers....on the 2t/t^3...I know division means subtraction...but can you walk me through that part please. Thank you for all your help.(0 votes)
How do we go from 1/3 x^3 to 4sqrtx^8?(2 votes)
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(2 votes)
It's so easy.
Just like differential calculus, integral calculus has its own rules.(2 votes)- how do you do this(2 votes)
- 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!(2 votes)
- how to integrate derivatives like (x+3)^4?(1 vote)
- What is the specific meaning of indefinite integral?(0 votes)
That depends on the application that you are using it in. In all scenarios it is the general antiderivative. For practical examples, if you take the integral of the velocity function, you get the displacement. If you take the integral of the absolute value of the velocity function (aka speed function), you get the distance traveled.(2 votes)
I want anyone to tell me about standard integral(0 votes)
Standard integral is a table of basic formulas used to find the antiderivative of given function(2 votes)