You can access our Handy Table of Integrals from the Reference menu at the top of the screen at any time.

Summary: Definite Integration

You can practice each of these common integration problems below.

I. Power Rule

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
\[\bbox[yellow,5px]{
\begin{align*}
\int_a^b \! x^n\, dx &= \frac{1}{n+1}\big[ x^{n+1}\big]_a^b \qquad (n \ne -1) \\[8px]
&= \frac{1}{n+1}\left[b^{n+1} - a^{n+1} \right]
\end{align*}} \]
For example,
\[ \begin{align*}
\int_a^b \! x^3\, dx &= \frac{1}{3+1}\big[ x^{3+1}\big]_a^b \qquad \phantom{(n \ne -1) } \\[8px]
&= \frac{1}{4}\big[ x^{4}\big]_a^b \\[8px]
&= \frac{1}{4} \left[b^4 - a^4 \right]
\end{align*} \]

Power Rule Problem #1

Power Rule Problem #2

Power Rule Problem #3

Evaluate the integral: $\displaystyle{\int_{-3}^0 \! \left(\frac{2}{3}w^5 - \frac{1}{4}w^3 + w \right) dw.}$

Power Rule Problem #4

Evaluate the integral: $\displaystyle{\int_5^0 \! \left( x^2 -1\right)^2 \, dx . }$

Power Rule Problem #5

Evaluate the integral: $\displaystyle{ \int_1^2 \!\frac{4x - 7x^5}{x^3} \, dx. }$

Power Rule Problem #6

Power Rule Problem #7

Evaluate the integral: $\displaystyle{\int_4^9 \! \frac{x -5}{\sqrt{x}} \, dx.}$

Power Rule Problem #8

Evaluate the integral: $\displaystyle{\int_0^1 \!\left( 5 \sqrt{x^5} + 2 \sqrt[3]{x^2} \right) \, dx .}$

II. Exponential Rule: e^x

This integral is the easiest to remember: since $(e^x)' = e^x$, the integral of $e^x$ is also $e^x$
\[ \bbox[yellow,5px]{
\begin{align*}
\int_a^b \! e^x \, dx &= \big[ e^x\big]_a^b \\[8px]
&= e^b - e^a
\end{align*}} \]

Exponential Rule e^x Problem #1

Evaluate the integral: $\displaystyle{\int_1^3 \! 5e^x \, dx .}$

Exponential Rule e^x Problem #2

Evaluate the integral: $\displaystyle{\int_1^4 \! \left( \frac{1}{\sqrt{x^5}} - e^x\right) \, dx .}$

III. Exponential Rule: c^x

Here $c$ is a constant (a number, any number). Then
\[\bbox[yellow,5px]{
\begin{align*}
\int_a^b \! c^x &= \frac{1}{\ln c} \big[ c^x \big]_a^b \\[8px]
&= \frac{1}{\ln c} \big[c^b - c^a \big]
\end{align*}
}
\]

Open to see how this rule encompasses the e^x rule

Recall that $\ln e = 1$. Then
\begin{align*}
\int_a^b \! e^x &= \frac{1}{\ln e} \big[ e^x \big]_a^b \\[8px]
&= \big[ e^x \big]_a^b = \big[e^b – e^a \big]
\end{align*}

[collapse]

Exponential Rule c^x Problem #1

Evaluate the integral: $\displaystyle{\int_0^2 \! 3^x \, dx .}$

Exponential Rule c^x Problem #2

Evaluate the integral: $\displaystyle{\int_0^1 \! \left( x^5 + 5^x\right) \, dx }.$

IV. Trigonometric Rules

We're listing here only the trig integrals that you should be familiar with at this early stage: each of these follows directly from a derivative you immediately know. For example, since
$$(\sin x)' = \cos x$$
we know immediately that
$$\int_a^b \! \cos x \, dx = \big[ \sin x \big]_a^b$$
Accordingly: \[ \bbox[yellow,5px]{\begin{align*}
\int_a^b \cos x \, dx &= \big[\sin x\big]_a^b \\[8px]
\int_a^b \sin x \, dx &= -\big[\cos x\big]_a^b \\[8px]
\int_a^b \sec^2 x \, dx &= \big[\tan x\big]_a^b \\[8px]
\int_a^b \sec x \tan x \, dx &= \big[\sec x\big]_a^b \\[8px]
\int_a^b \csc^2 x \, dx &= -\big[\cot x\big]_a^b \\[8px]
\end{align*}} \]
Each integral follows directly from a derivative you know. You can review those using our Trig Function Derivatives table; it's always available from the Reference menu at the top of every page.

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: \[ \bbox[10px,border:2px solid blue]{ \begin{align*} \sin^2 x &= \frac{1}{2} - \frac{1}{2}\cos 2x \\ \\ \cos^2 x &= \frac{1}{2} + \frac{1}{2}\cos 2x \end{align*}}\] Then \begin{align*} \int_a^b \sin^2 x \, dx &= \int \left(\frac{1}{2} - \frac{1}{2}\cos 2x \right) \, dx = \left[\frac{1}{2}x - \frac{1}{4}\sin 2x\right]_a^b \\[8px] \int_a^b \cos^2 x \, dx &= \int \left( \frac{1}{2} + \frac{1}{2}\cos 2x \right) \, dx = \left[\frac{1}{2}x + \frac{1}{4}\sin 2x\right]_a^b \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: \[ \bbox[10px,border:2px solid blue]{ \begin{align*} \sin^2 x &= \frac{1}{2} - \frac{1}{2}\cos 2x \\ \\ \cos^2 x &= \frac{1}{2} + \frac{1}{2}\cos 2x \end{align*}}\] Then \begin{align*} \int_a^b \sin^2 x \, dx &= \int \left(\frac{1}{2} - \frac{1}{2}\cos 2x \right) \, dx = \left[\frac{1}{2}x - \frac{1}{4}\sin 2x\right]_a^b \\[8px] \int_a^b \cos^2 x \, dx &= \int \left( \frac{1}{2} + \frac{1}{2}\cos 2x \right) \, dx = \left[\frac{1}{2}x + \frac{1}{4}\sin 2x\right]_a^b \end{align*}

Trig Problem #1

Evaluate the integral: $\displaystyle{ \int_0^{\pi/2} \!4 \cos x \, dx .}$

Trig Problem #2

Evaluate the integral: $\displaystyle{\int_0^{\pi} \!\frac{1}{2}\, \sin \theta \, d\theta .}$

Trig Problem #3

Evaluate the integral: $\displaystyle{\int_0^{\pi/4} \frac{dx}{\sec x} .}$

Trig Problem #4

Evaluate the integral: $\displaystyle{\int_0^{\pi/4} \! \sec^2 x \, dx .}$

Trig Problem #5

Evaluate the integral: $\displaystyle{ \int_{\pi/6}^{\pi/3} \! \sec x \tan x \, dx. }$

Trig Problem #6

Evaluate the integral: $\displaystyle{\int_{\pi/4}^{3\pi/4} \! \left(\csc^2 x + 2x^2 \right) \, dx . }$

Trig Problem #7

Evaluate the integral: $\displaystyle{\int_0^{\pi/3} \! \frac{3 + 2\cos^2 x}{\cos^2 x} \, dx . }$

You may also be expected to use the Trig Identity and its variants:
\[ \bbox[10px,border:2px solid blue]{
\begin{align*}
\sin^2 x + \cos^2 x &= 1 \\[8px]
1 + \cot^2 x &= \csc^2 x \\[8px]
1 + \tan^2 x &= \sec^2 x
\end{align*}}\]
We'll of course illustrate the use of these identities in the problems below.

Trig Problem #8

Evaluate the integral: $\displaystyle{ \int_0^{\pi/4} \! \tan^2 x \, dx .}$

Trig Problem #9

Evaluate the integral: $\displaystyle{\int_0^{\pi/4} \!\frac{5 \sin x + 5 \sin x \tan^2 x}{\sec^2 x} \, dx. }$

Trig Problem #10

Evaluate the integral: $\displaystyle{\int_{\pi/6}^{\pi/3} \! \frac{\sec^3 x - \sec x}{\tan x} \, dx .}$

We'd love to hear:

- What questions do you have about the solutions above?
- Which ones are giving you the most trouble?
- What other integration problems are you trying to work through for your class?

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