Calculus Examples

$\int (x + 2) \sqrt{x - 2} d x$

Step 1

Let $u = x - 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $x - 2$ .

$\frac{d}{d x} [x - 2]$

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

$\int (u + 2 + 2) \sqrt{u} d u$

Step 2

Simplify the expression.

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Step 2.1

Add $2$ and $2$ .

$\int (u + 4) \sqrt{u} d u$

Step 2.2

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

$\int (u + 4) u^{\frac{1}{2}} d u$

Step 3

Expand $(u + 4) u^{\frac{1}{2}}$ .

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Step 3.1

Apply the distributive property.

$\int u \cdot u^{\frac{1}{2}} + 4 u^{\frac{1}{2}} d u$

Step 3.2

Raise $u$ to the power of $1$ .

$\int u^{1} u^{\frac{1}{2}} + 4 u^{\frac{1}{2}} d u$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u^{1 + \frac{1}{2}} + 4 u^{\frac{1}{2}} d u$

Step 3.4

Write $1$ as a fraction with a common denominator.

$\int u^{\frac{2}{2} + \frac{1}{2}} + 4 u^{\frac{1}{2}} d u$

Step 3.5

Combine the numerators over the common denominator.

$\int u^{\frac{2 + 1}{2}} + 4 u^{\frac{1}{2}} d u$

Step 3.6

Add $2$ and $1$ .

$\int u^{\frac{3}{2}} + 4 u^{\frac{1}{2}} d u$

Step 3.7

Reorder $u^{\frac{3}{2}}$ and $4 u^{\frac{1}{2}}$ .

$\int 4 u^{\frac{1}{2}} + u^{\frac{3}{2}} d u$

Step 4

Split the single integral into multiple integrals.

$\int 4 u^{\frac{1}{2}} d u + \int u^{\frac{3}{2}} d u$

Step 5

Since $4$ is constant with respect to $u$ , move $4$ out of the integral.

$4 \int u^{\frac{1}{2}} d u + \int u^{\frac{3}{2}} d u$

Step 6

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

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

Step 7

By the Power Rule, the integral of $u^{\frac{3}{2}}$ with respect to $u$ is $\frac{2}{5} u^{\frac{5}{2}}$ .

$4 (\frac{2}{3} u^{\frac{3}{2}} + C) + \frac{2}{5} u^{\frac{5}{2}} + C$

Step 8

Simplify.

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Step 8.1

Combine $\frac{2}{3}$ and $u^{\frac{3}{2}}$ .

$4 (\frac{2 u^{\frac{3}{2}}}{3} + C) + \frac{2}{5} u^{\frac{5}{2}} + C$

Step 8.2

Simplify.

$\frac{8 u^{\frac{3}{2}}}{3} + \frac{2}{5} u^{\frac{5}{2}} + C$

Step 8.3

Reorder terms.

$\frac{8}{3} u^{\frac{3}{2}} + \frac{2}{5} u^{\frac{5}{2}} + C$

Step 9

Replace all occurrences of $u$ with $x - 2$ .

$\frac{8}{3} {(x - 2)}^{\frac{3}{2}} + \frac{2}{5} {(x - 2)}^{\frac{5}{2}} + C$