Calculus Examples

$\int 10.8 \sqrt{- 4 x + 5} d x$

Step 1

Since $10.8$ is constant with respect to $x$ , move $10.8$ out of the integral.

$10.8 \int \sqrt{- 4 x + 5} d x$

Step 2

Let $u = - 4 x + 5$ . Then $d u = - 4 d x$ , so $- \frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = - 4 x + 5$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $- 4 x + 5$ .

$\frac{d}{d x} [- 4 x + 5]$

Step 2.1.2

By the Sum Rule, the derivative of $- 4 x + 5$ with respect to $x$ is $\frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 4 x]$ .

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Step 2.1.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$- 4 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 2.1.3.2

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

$- 4 \cdot 1 + \frac{d}{d x} [5]$

Step 2.1.3.3

Multiply $- 4$ by $1$ .

$- 4 + \frac{d}{d x} [5]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

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

$- 4 + 0$

Step 2.1.4.2

Add $- 4$ and $0$ .

$- 4$

Step 2.2

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

$10.8 \int \sqrt{u} \frac{1}{- 4} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$10.8 \int \sqrt{u} (- \frac{1}{4}) d u$

Step 3.2

Combine $\sqrt{u}$ and $\frac{1}{4}$ .

$10.8 \int - \frac{\sqrt{u}}{4} d u$

Step 4

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

$10.8 (- \int \frac{\sqrt{u}}{4} d u)$

Step 5

Multiply $- 1$ by $10.8$ .

$- 10.8 \int \frac{\sqrt{u}}{4} d u$

Step 6

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$- 10.8 (\frac{1}{4} \int \sqrt{u} d u)$

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

Combine $\frac{1}{4}$ and $- 10.8$ .

$\frac{- 10.8}{4} \int \sqrt{u} d u$

Step 7.1.2

Move the negative in front of the fraction.

$- \frac{10.8}{4} \int \sqrt{u} d u$

Step 7.2

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

$- \frac{10.8}{4} \int u^{\frac{1}{2}} d u$

Step 8

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

$- \frac{10.8}{4} (\frac{2}{3} u^{\frac{3}{2}} + C)$

Step 9

Simplify.

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Step 9.1

Rewrite $- \frac{10.8}{4} (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $- 1 \cdot 2.7 (\frac{2}{3} u^{\frac{3}{2}}) + C$ .

$- 1 \cdot 2.7 (\frac{2}{3} u^{\frac{3}{2}}) + C$

Step 9.2

Simplify.

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Step 9.2.1

Multiply $- 1$ by $2.7$ .

$- 2.7 (\frac{2}{3} u^{\frac{3}{2}}) + C$

Step 9.2.2

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

$- 2.7 \frac{2 u^{\frac{3}{2}}}{3} + C$

Step 9.2.3

Combine $- 2.7$ and $\frac{2 u^{\frac{3}{2}}}{3}$ .

$\frac{- 2.7 (2 u^{\frac{3}{2}})}{3} + C$

Step 9.2.4

Multiply $2$ by $- 2.7$ .

$\frac{- 5.4 u^{\frac{3}{2}}}{3} + C$

Step 9.2.5

Move the negative in front of the fraction.

$- \frac{5.4 u^{\frac{3}{2}}}{3} + C$

Step 10

Replace all occurrences of $u$ with $- 4 x + 5$ .

$- \frac{5.4 {(- 4 x + 5)}^{\frac{3}{2}}}{3} + C$

Step 11

Reorder terms.

$- \frac{5.4}{3} {(- 4 x + 5)}^{\frac{3}{2}} + C$