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Integration, Definite

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Summary: Definite Integration
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Important Notation

When we write $\big[F(x)\big]_a^b$, it means compute $F(b)$ and then subtract $F(a)$:
$$\big[F(x)\big]_a^b = F(b) – F(a)$$

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_a^b \! x^n\, dx &= \frac{1}{n+1}\big[ x^{n+1}\big]_a^b \qquad (n \ne -1) \\[8px]
&= \frac{1}{n+1}\left[b^{n+1} – a^{n+1} \right]

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 $e^x$
\[ \begin{align*}
\int_a^b \! e^x \, dx &= \big[ e^x\big]_a^b \\[8px]
&= e^b – e^a
\end{align*} \]

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

\int_a^b \! c^x &= \frac{1}{\ln c} \big[ c^x \big]_a^b \\[8px]
&= \frac{1}{\ln c} \big[c^b – c^a \big]

Trigonometric Rules: Integrals of Trig Functions

\[ \begin{align*}
\int_a^b \cos x \, dx &= \big[\sin x\big]_a^b \\[8px]
\int_a^b \sin x \, dx &= -\big[\cos x\big]_a^b \\[8px]
\int_a^b \sec^2 x \, dx &= \big[\tan x\big]_a^b \\[8px]
\int_a^b \sec x \tan x \, dx &= \big[\sec x\big]_a^b \\[8px]
\int_a^b \csc^2 x \, dx &= -\big[\cot x\big]_a^b \\[8px]
\end{align*} \]

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:

\int_a^b \sin^2 x \, dx &= \int_a^b \left(\frac{1}{2} – \frac{1}{2}\cos 2x \right) \, dx = \big[\frac{1}{2}x – \frac{1}{4}\sin 2x \big]_a^b \\[8px]
\int_a^b \cos^2 x \, dx &= \int_a^b \left( \frac{1}{2} + \frac{1}{2}\cos 2x \right) \, dx = \big[\frac{1}{2}x + \frac{1}{4}\sin 2x \big]_a^b


You can practice each of these common integration problems below.

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