Calculus Examples

$\int (\sqrt[3]{x} + \cos (2 x + 5)) d x$

Step 1

Remove parentheses.

$\int \sqrt[3]{x} + \cos (2 x + 5) d x$

Step 2

Split the single integral into multiple integrals.

$\int \sqrt[3]{x} d x + \int \cos (2 x + 5) d x$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\int x^{\frac{1}{3}} d x + \int \cos (2 x + 5) d x$

Step 4

By the Power Rule, the integral of $x^{\frac{1}{3}}$ with respect to $x$ is $\frac{3}{4} x^{\frac{4}{3}}$ .

$\frac{3}{4} x^{\frac{4}{3}} + C + \int \cos (2 x + 5) d x$

Step 5

Let $u = 2 x + 5$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 2 x + 5$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $2 x + 5$ .

$\frac{d}{d x} [2 x + 5]$

Step 5.1.2

By the Sum Rule, the derivative of $2 x + 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [5]$

Step 5.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 5.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 5.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [5]$

Step 5.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [5]$

Step 5.1.4

Differentiate using the Constant Rule.

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Step 5.1.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$2 + 0$

Step 5.1.4.2

Add $2$ and $0$ .

$2$

Step 5.2

Rewrite the problem using $u$ and $d u$ .

$\frac{3}{4} x^{\frac{4}{3}} + C + \int \cos (u) \frac{1}{2} d u$

Step 6

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\frac{3}{4} x^{\frac{4}{3}} + C + \int \frac{\cos (u)}{2} d u$

Step 7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{3}{4} x^{\frac{4}{3}} + C + \frac{1}{2} \int \cos (u) d u$

Step 8

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{3}{4} x^{\frac{4}{3}} + C + \frac{1}{2} (\sin (u) + C)$

Step 9

Simplify.

$\frac{3}{4} x^{\frac{4}{3}} + \frac{1}{2} \sin (u) + C$

Step 10

Replace all occurrences of $u$ with $2 x + 5$ .

$\frac{3}{4} x^{\frac{4}{3}} + \frac{1}{2} \sin (2 x + 5) + C$