Calculus Examples

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

Step 1

Let $u = x - 5$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $x - 5$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ 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 $d u$ .

$\int 2 (u + 5) \sqrt{u} d u$

Step 2

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

$2 \int (u + 5) \sqrt{u} d u$

Step 3

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

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

Step 4

Expand $(u + 5) u^{\frac{1}{2}}$ .

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Add $2$ and $1$ .

$2 \int u^{\frac{3}{2}} + 5 u^{\frac{1}{2}} d u$

Step 4.7

Reorder $u^{\frac{3}{2}}$ and $5 u^{\frac{1}{2}}$ .

$2 \int 5 u^{\frac{1}{2}} + u^{\frac{3}{2}} d u$

Step 5

Split the single integral into multiple integrals.

$2 (\int 5 u^{\frac{1}{2}} d u + \int u^{\frac{3}{2}} d u)$

Step 6

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

$2 (5 \int u^{\frac{1}{2}} d u + \int u^{\frac{3}{2}} d u)$

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

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

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

Step 9.1.2

Combine $\frac{2}{5}$ and $u^{\frac{5}{2}}$ .

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

Step 9.2

Simplify.

$2 (\frac{10 u^{\frac{3}{2}}}{3} + \frac{2 u^{\frac{5}{2}}}{5}) + C$

Step 9.3

Reorder terms.

$2 (\frac{10}{3} u^{\frac{3}{2}} + \frac{2}{5} u^{\frac{5}{2}}) + C$

Step 10

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

$2 (\frac{10}{3} {(x - 5)}^{\frac{3}{2}} + \frac{2}{5} {(x - 5)}^{\frac{5}{2}}) + C$

Step 11

Simplify.

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Step 11.1

Simplify each term.

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Step 11.1.1

Combine $\frac{10}{3}$ and ${(x - 5)}^{\frac{3}{2}}$ .

$2 (\frac{10 {(x - 5)}^{\frac{3}{2}}}{3} + \frac{2}{5} {(x - 5)}^{\frac{5}{2}}) + C$

Step 11.1.2

Combine $\frac{2}{5}$ and ${(x - 5)}^{\frac{5}{2}}$ .

$2 (\frac{10 {(x - 5)}^{\frac{3}{2}}}{3} + \frac{2 {(x - 5)}^{\frac{5}{2}}}{5}) + C$

Step 11.2

To write $\frac{10 {(x - 5)}^{\frac{3}{2}}}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2 (\frac{10 {(x - 5)}^{\frac{3}{2}}}{3} \cdot \frac{5}{5} + \frac{2 {(x - 5)}^{\frac{5}{2}}}{5}) + C$

Step 11.3

To write $\frac{2 {(x - 5)}^{\frac{5}{2}}}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 (\frac{10 {(x - 5)}^{\frac{3}{2}}}{3} \cdot \frac{5}{5} + \frac{2 {(x - 5)}^{\frac{5}{2}}}{5} \cdot \frac{3}{3}) + C$

Step 11.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.4.1

Multiply $\frac{10 {(x - 5)}^{\frac{3}{2}}}{3}$ by $\frac{5}{5}$ .

$2 (\frac{10 {(x - 5)}^{\frac{3}{2}} \cdot 5}{3 \cdot 5} + \frac{2 {(x - 5)}^{\frac{5}{2}}}{5} \cdot \frac{3}{3}) + C$

Step 11.4.2

Multiply $3$ by $5$ .

$2 (\frac{10 {(x - 5)}^{\frac{3}{2}} \cdot 5}{15} + \frac{2 {(x - 5)}^{\frac{5}{2}}}{5} \cdot \frac{3}{3}) + C$

Step 11.4.3

Multiply $\frac{2 {(x - 5)}^{\frac{5}{2}}}{5}$ by $\frac{3}{3}$ .

$2 (\frac{10 {(x - 5)}^{\frac{3}{2}} \cdot 5}{15} + \frac{2 {(x - 5)}^{\frac{5}{2}} \cdot 3}{5 \cdot 3}) + C$

Step 11.4.4

Multiply $5$ by $3$ .

$2 (\frac{10 {(x - 5)}^{\frac{3}{2}} \cdot 5}{15} + \frac{2 {(x - 5)}^{\frac{5}{2}} \cdot 3}{15}) + C$

Step 11.5

Combine the numerators over the common denominator.

$2 \frac{10 {(x - 5)}^{\frac{3}{2}} \cdot 5 + 2 {(x - 5)}^{\frac{5}{2}} \cdot 3}{15} + C$

Step 11.6

Simplify the numerator.

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Step 11.6.1

Factor $2 {(x - 5)}^{\frac{3}{2}}$ out of $10 {(x - 5)}^{\frac{3}{2}} \cdot 5 + 2 {(x - 5)}^{\frac{5}{2}} \cdot 3$ .

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Step 11.6.1.1

Reorder the expression.

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Step 11.6.1.1.1

Move ${(x - 5)}^{\frac{3}{2}}$ .

$2 \frac{10 \cdot 5 {(x - 5)}^{\frac{3}{2}} + 2 {(x - 5)}^{\frac{5}{2}} \cdot 3}{15} + C$

Step 11.6.1.1.2

Move ${(x - 5)}^{\frac{5}{2}}$ .

$2 \frac{10 \cdot 5 {(x - 5)}^{\frac{3}{2}} + 2 \cdot 3 {(x - 5)}^{\frac{5}{2}}}{15} + C$

Step 11.6.1.2

Factor $2 {(x - 5)}^{\frac{3}{2}}$ out of $10 \cdot 5 {(x - 5)}^{\frac{3}{2}}$ .

$2 \frac{2 {(x - 5)}^{\frac{3}{2}} (5 \cdot 5) + 2 \cdot 3 {(x - 5)}^{\frac{5}{2}}}{15} + C$

Step 11.6.1.3

Factor $2 {(x - 5)}^{\frac{3}{2}}$ out of $2 \cdot 3 {(x - 5)}^{\frac{5}{2}}$ .

$2 \frac{2 {(x - 5)}^{\frac{3}{2}} (5 \cdot 5) + 2 {(x - 5)}^{\frac{3}{2}} (3 {(x - 5)}^{\frac{2}{2}})}{15} + C$

Step 11.6.1.4

Factor $2 {(x - 5)}^{\frac{3}{2}}$ out of $2 {(x - 5)}^{\frac{3}{2}} (5 \cdot 5) + 2 {(x - 5)}^{\frac{3}{2}} (3 {(x - 5)}^{\frac{2}{2}})$ .

$2 \frac{2 {(x - 5)}^{\frac{3}{2}} (5 \cdot 5 + 3 {(x - 5)}^{\frac{2}{2}})}{15} + C$

Step 11.6.2

Multiply $5$ by $5$ .

$2 \frac{2 {(x - 5)}^{\frac{3}{2}} (25 + 3 {(x - 5)}^{\frac{2}{2}})}{15} + C$

Step 11.6.3

Simplify each term.

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Step 11.6.3.1

Divide $2$ by $2$ .

$2 \frac{2 {(x - 5)}^{\frac{3}{2}} (25 + 3 {(x - 5)}^{1})}{15} + C$

Step 11.6.3.2

Simplify.

$2 \frac{2 {(x - 5)}^{\frac{3}{2}} (25 + 3 (x - 5))}{15} + C$

Step 11.6.3.3

Apply the distributive property.

$2 \frac{2 {(x - 5)}^{\frac{3}{2}} (25 + 3 x + 3 \cdot - 5)}{15} + C$

Step 11.6.3.4

Multiply $3$ by $- 5$ .

$2 \frac{2 {(x - 5)}^{\frac{3}{2}} (25 + 3 x - 15)}{15} + C$

Step 11.6.4

Subtract $15$ from $25$ .

$2 \frac{2 {(x - 5)}^{\frac{3}{2}} (3 x + 10)}{15} + C$

Step 11.7

Multiply $2 \frac{2 {(x - 5)}^{\frac{3}{2}} (3 x + 10)}{15}$ .

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Step 11.7.1

Combine $2$ and $\frac{2 {(x - 5)}^{\frac{3}{2}} (3 x + 10)}{15}$ .

$\frac{2 (2 {(x - 5)}^{\frac{3}{2}} (3 x + 10))}{15} + C$

Step 11.7.2

Multiply $2$ by $2$ .

$\frac{4 ({(x - 5)}^{\frac{3}{2}} (3 x + 10))}{15} + C$

$\frac{4 {(x - 5)}^{\frac{3}{2}} (3 x + 10)}{15} + C$