Calculus Examples

$5 \sqrt{x - 3}$

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \sqrt{x - 3} d x$

Step 2

Let $u = x - 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = x - 3$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x - 3$ .

$\frac{d}{d x} [x - 3]$

Step 2.1.2

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 2.1.3

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

$1 + \frac{d}{d x} [- 3]$

Step 2.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

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

$5 \int \sqrt{u} d u$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$5 \int u^{\frac{1}{2}} d u$

Step 4

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$5 (\frac{2}{3} u^{\frac{3}{2}} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $5 (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $5 (\frac{2}{3}) u^{\frac{3}{2}} + C$ .

$5 (\frac{2}{3}) u^{\frac{3}{2}} + C$

Step 5.2

Simplify.

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Step 5.2.1

Combine $5$ and $\frac{2}{3}$ .

$\frac{5 \cdot 2}{3} u^{\frac{3}{2}} + C$

Step 5.2.2

Multiply $5$ by $2$ .

$\frac{10}{3} u^{\frac{3}{2}} + C$

Step 6

Replace all occurrences of $u$ with $x - 3$ .

$\frac{10}{3} {(x - 3)}^{\frac{3}{2}} + C$