Free practice problems, each with a complete solution, of computing typical beginning definite integrals that use the power rule, exponentials, and basic trig functions.

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

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