Calculus Examples

$\sqrt{1 + x}$

Step 1

Write $\sqrt{1 + x}$ as a function.

$f (x) = \sqrt{1 + x}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \sqrt{1 + x} d x$

Step 4

Let $u = 1 + x$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 1 + x$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $1 + x$ .

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

Step 4.1.2

By the Sum Rule, the derivative of $1 + x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x]$ .

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

Step 4.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x]$

Step 4.1.4

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

$0 + 1$

Step 4.1.5

Add $0$ and $1$ .

$1$

Step 4.2

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

$\int \sqrt{u} d u$

Step 5

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

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

Step 6

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

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

Step 7

Replace all occurrences of $u$ with $1 + x$ .

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

Step 8

The answer is the antiderivative of the function $f (x) = \sqrt{1 + x}$ .

$F (x) =$ $\frac{2}{3} {(1 + x)}^{\frac{3}{2}} + C$