Free practice problems, each with a complete solution, for typical beginning indefinite integrals in Calculus.

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

Summary: Indefinite Integration

You can of course practice typical problems below, each with a complete solution.

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. Finally add *C*." We can express this process mathematically as
\[\bbox[yellow,5px]{
\int \! x^n\, dx = \left(\frac{1}{n+1} \right)x^{n+1} + C \qquad (n \ne -1)
}\]
For example,
\[ \begin{align*}
\int \! x^3\, dx &= \left(\frac{1}{3+1} \right)x^{3+1} + C \qquad \phantom{(n \ne -1) } \\[8px]
&= \frac{1}{4}x^{4} + C \\[8px]
\end{align*} \]

Power Rule Problem #1

Power Rule Problem #2

Power Rule Problem #3

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

Power Rule Problem #4

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

Power Rule Problem #5

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

Power Rule Problem #6

Power Rule Problem #7

Find: $\displaystyle{\int \! \frac{x -5}{\sqrt{x}} \, dx.}$

Power Rule Problem #8

Find $\displaystyle{\int \!\left( 5 \sqrt{x^5} + 2 \sqrt[3]{x^2} \right) \, dx .}$

II. Exponential Rule: e^x

This integral is the easiest to remember: since $(e^x)' = e^x$, the integral of $e^x$ is also $e^x:$
\[ \bbox[yellow,5px]{
\int \! e^x \, dx = e^x + C } \]

Exponential Rule e^x Problem #1

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

Exponential Rule e^x Problem #2

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

III. Exponential Rule: k^x

\[\bbox[yellow,5px]{
\int \! k^x = \frac{1}{\ln k} k^x + C} \]

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

Recall that $\ln e = 1$. Then
\begin{align*}
\int \! e^x &= \frac{1}{\ln e} e^x + C \\[8px]
&= e^x + C
\end{align*}

[collapse]

Exponential Rule c^x Problem #1

Find: $\displaystyle{\int \! 3^x \, dx .}$

Exponential Rule c^x Problem #2

Find: $\displaystyle{\int \! \left( x^5 + 5^x\right) \, dx }.$

IV. Trigonometric Rules

We're listing here on the trig integrals that you should know at this early stage because each follows directly from a derivative you know. For example, since
$$(\sin x)' = \cos x$$
we know immediately that
$$\int \! \cos x \, dx = \sin x + C $$

Accordingly: \[ \bbox[yellow,5px]{\begin{align*} \int \cos x \, dx &= \sin x + C \\[8px] \int \sin x \, dx &= -\cos x + C \\[8px] \int \sec^2 x \, dx &= \tan x + C \\[8px] \int \sec x \tan x \, dx &= \sec x + C \\[8px] \int \csc^2 x \, dx &= -\cot x + C \\[8px] \end{align*}} \] 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: \[ \bbox[10px,border:2px solid blue]{ \begin{align*} \sin^2 x &= \frac{1}{2} - \frac{1}{2}\cos 2x \\ \\ \cos^2 x &= \frac{1}{2} + \frac{1}{2}\cos 2x \end{align*}}\] Then \begin{align*} \int \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 \quad [*] \\[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 \quad [*] \\[8px] \end{align*}

Accordingly: \[ \bbox[yellow,5px]{\begin{align*} \int \cos x \, dx &= \sin x + C \\[8px] \int \sin x \, dx &= -\cos x + C \\[8px] \int \sec^2 x \, dx &= \tan x + C \\[8px] \int \sec x \tan x \, dx &= \sec x + C \\[8px] \int \csc^2 x \, dx &= -\cot x + C \\[8px] \end{align*}} \] 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: \[ \bbox[10px,border:2px solid blue]{ \begin{align*} \sin^2 x &= \frac{1}{2} - \frac{1}{2}\cos 2x \\ \\ \cos^2 x &= \frac{1}{2} + \frac{1}{2}\cos 2x \end{align*}}\] Then \begin{align*} \int \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 \quad [*] \\[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 \quad [*] \\[8px] \end{align*}

Trig Problem #1

Find: $\displaystyle{ \int \!4 \cos x \, dx .}$

Trig Problem #2

Find: $\displaystyle{\int \!\frac{1}{2}\, \sin \theta \, d\theta .}$

Trig Problem #3

Find: $\displaystyle{\int \frac{dx}{\sec x} .}$

Trig Problem #4

Find: $\displaystyle{\int \! \sec^2 x \, dx .}$

Trig Problem #5

Find: $\displaystyle{ \int \! \sec x \tan x \, dx. }$

Trig Problem #6

Find: $\displaystyle{\int \! \left(\csc^2 x + 2x^2 \right) \, dx . }$

Trig Problem #7

Find: $\displaystyle{\int \! \frac{3 + 2\cos^2 x}{\cos^2 x} \, dx . }$

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

Find: $\displaystyle{ \int \! \tan^2 x \, dx .}$

Trig Problem #9

Find: $\displaystyle{\int \!\frac{5 \sin x + 5 \sin x \tan^2 x}{\sec^2 x} \, dx. }$

Trig Problem #10

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

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