Calculus Examples

$2 (\sqrt{- 3 x + 5}) + 3$

Step 1

Split the single integral into multiple integrals.

$\int 2 \sqrt{- 3 x + 5} d x + \int 3 d x$

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \sqrt{- 3 x + 5} d x + \int 3 d x$

Step 3

Let $u = - 3 x + 5$ . Then $d u = - 3 d x$ , so $- \frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = - 3 x + 5$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $- 3 x + 5$ .

$\frac{d}{d x} [- 3 x + 5]$

Step 3.1.2

By the Sum Rule, the derivative of $- 3 x + 5$ with respect to $x$ is $\frac{d}{d x} [- 3 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [- 3 x] + \frac{d}{d x} [5]$

Step 3.1.3

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

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Step 3.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$- 3 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 3.1.3.2

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

$- 3 \cdot 1 + \frac{d}{d x} [5]$

Step 3.1.3.3

Multiply $- 3$ by $1$ .

$- 3 + \frac{d}{d x} [5]$

Step 3.1.4

Differentiate using the Constant Rule.

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Step 3.1.4.1

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

$- 3 + 0$

Step 3.1.4.2

Add $- 3$ and $0$ .

$- 3$

Step 3.2

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

$2 \int \sqrt{u} \frac{1}{- 3} d u + \int 3 d x$

Step 4

Simplify.

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Step 4.1

Move the negative in front of the fraction.

$2 \int \sqrt{u} (- \frac{1}{3}) d u + \int 3 d x$

Step 4.2

Combine $\sqrt{u}$ and $\frac{1}{3}$ .

$2 \int - \frac{\sqrt{u}}{3} d u + \int 3 d x$

Step 5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$2 (- \int \frac{\sqrt{u}}{3} d u) + \int 3 d x$

Step 6

Multiply $- 1$ by $2$ .

$- 2 \int \frac{\sqrt{u}}{3} d u + \int 3 d x$

Step 7

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$- 2 (\frac{1}{3} \int \sqrt{u} d u) + \int 3 d x$

Step 8

Simplify the expression.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{3}$ and $- 2$ .

$\frac{- 2}{3} \int \sqrt{u} d u + \int 3 d x$

Step 8.1.2

Move the negative in front of the fraction.

$- \frac{2}{3} \int \sqrt{u} d u + \int 3 d x$

Step 8.2

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

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

Step 9

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

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

Step 10

Apply the constant rule.

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

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Simplify.

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Step 11.2.1

Multiply $\frac{2}{3}$ by $\frac{2}{3}$ .

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

Step 11.2.2

Multiply $2$ by $2$ .

$- \frac{4}{3 \cdot 3} u^{\frac{3}{2}} + 3 x + C$

Step 11.2.3

Multiply $3$ by $3$ .

$- \frac{4}{9} u^{\frac{3}{2}} + 3 x + C$

Step 12

Replace all occurrences of $u$ with $- 3 x + 5$ .

$- \frac{4}{9} {(- 3 x + 5)}^{\frac{3}{2}} + 3 x + C$