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Integration, Indefinite

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Summary: Indefinite Integration
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Power Rule: Integration of $x^n$ $(n \ne -1)$

If you’re integrating x-to-some-power (except $x^{-1}$), the rule to remember is: “Increase the power by 1, and then divide by the new power.” We can express this process mathematically as
\[\int \! x^n\, dx = \frac{1}{n+1} x^{n+1} + C \qquad (n \ne -1) \]


Exponential Rule: Integration of $e^x$

This integral is the easiest to remember: since $(e^x)’ = e^x$, the integral of $e^x$ is also $e^x$
\[\int \! e^x \, dx = e^x+ C \]


Exponential Rule: Integration of $k^x$ (where $k$ is a constant)

\[ \int \! k^x = \frac{1}{\ln k} k^x + C \]


Trigonometric Rules: Integrals of Trig Functions

\[ \begin{align*}
\int \cos x \, dx &= \sin x + C \\[8px]
\int \sin x \, dx &= -\cos x + C \\[8px]
\int \sec^2 x \, dx &= \tan x + C \\[8px]
\int \sec x \tan x \, dx &= \sec x + C \\[8px]
\int \csc^2 x \, dx &= -\cot x + C \\[8px]
\end{align*} \]

You might also be expected to know the integrals for $\sin^2 x$ and $\cos^2 x$, because they follow immediately from use of the half-angle formulas:

\begin{align*}
\int \sin^2 x \, dx &= \int \left(\frac{1}{2} – \frac{1}{2}\cos 2x \right) \, dx = \frac{1}{2}x – \frac{1}{4}\sin 2x + C \\[8px]
\int \cos^2 x \, dx &= \int \left( \frac{1}{2} + \frac{1}{2}\cos 2x \right) \, dx = \frac{1}{2}x + \frac{1}{4}\sin 2x + C
\end{align*}

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