Part 05 · Integration · Card 2 of 10
Power rule for integrals
The integral of x^n dx is x^(n+1) / (n+1) + C, for n not equal to -1. Raise the power by one, then divide by the new power.
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Analogy
The power rule for derivatives, run backwards.
Example
Integral of 6x^2 - 4x + 1
6x^2 gives 6x^3/3 = 2x^3.-4x gives -4x^2/2 = -2x^2.1 gives x.Result: 2x^3 - 2x^2 + x + C.
In practice
Lets you integrate any polynomial one term at a time.
Practice
Flashcards
- State the power rule for integrals.The integral of x^n dx is x^(n+1)/(n+1) + C, for n not equal to -1.
Interview questions
- Evaluate the integral of 3x^2 + 2x from 1 to 2 and explain each step.
- Antiderivative is x^3 + x^2
- Evaluate at 2: 8 + 4 = 12
- Evaluate at 1: 1 + 1 = 2
- Subtract: 12 - 2 = 10
- Fundamental theorem of calculus justifies it
Questions people ask
Why does the power rule for integrals fail when n is -1?
Dividing by n + 1 would mean dividing by zero. The integral of 1 over x is a different function: the natural logarithm of absolute x, plus C. This is the one power of x that needs its own rule in the table.
How do I integrate roots and fractions?
Rewrite them as powers first. The square root of x is x to the one-half, and 1 over x squared is x to the minus two. Then apply the power rule: the integral of x to the one-half is two-thirds x to the three-halves.