Calculus / Unit 7: Antiderivatives and the Integral
7.2 Basic Integration Rules
Notes
Reverse the derivative rules.
- Power rule: $\displaystyle \int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.
- $\displaystyle \int \frac{1}{x}\,dx = \ln|x| + C$ and $\displaystyle \int e^x\,dx = e^x + C$.
- $\displaystyle \int \cos x\,dx = \sin x + C$ and $\displaystyle \int \sin x\,dx = -\cos x + C$.
- Constant multiples and sums pass through the integral.
- Example: $\displaystyle \int (6x^2 + 4x)\,dx = 2x^3 + 2x^2 + C$.
Practice
Find $\displaystyle \int (6x^2 + 4x)\,dx$.
Raise each power by 1 and divide by the new power. The last choice has no $+ C$.
Find $\displaystyle \int 2\sin x\,dx$.
An antiderivative of $\sin x$ is $-\cos x$.
Find $\displaystyle \int \frac{1}{x^2}\,dx$.
Write $x^{-2}$. Then $\displaystyle \frac{x^{-1}}{-1} = -\frac{1}{x}$.
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