Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $2 - x$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 1.1.3.2

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

$0 - 1 \cdot 1$

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 1.1.4

Subtract $1$ from $0$ .

$- 1$

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Factor out negative.

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 4.7

Add $2$ and $1$ .

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

Step 4.8

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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

Step 11

Reorder terms.

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

Simplify each term.

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Step 13.1.1

Combine $\frac{4}{3}$ and ${(2 - x)}^{\frac{3}{2}}$ .

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

Step 13.1.2

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

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

Step 13.1.3

Move $2$ to the left of ${(2 - x)}^{\frac{5}{2}}$ .

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

Step 13.2

To write $\frac{4 {(2 - x)}^{\frac{3}{2}}}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 13.3

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

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

Step 13.4

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

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Step 13.4.1

Multiply $\frac{4 {(2 - x)}^{\frac{3}{2}}}{3}$ by $\frac{5}{5}$ .

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

Step 13.4.2

Multiply $3$ by $5$ .

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

Step 13.4.3

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

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

Step 13.4.4

Multiply $5$ by $3$ .

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

Step 13.5

Combine the numerators over the common denominator.

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

Step 13.6

Simplify the numerator.

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Step 13.6.1

Multiply $5$ by $4$ .

$- \frac{20 {(2 - x)}^{\frac{3}{2}} - 2 {(2 - x)}^{\frac{5}{2}} \cdot 3}{15} + C$

Step 13.6.2

Multiply $3$ by $- 2$ .

$- \frac{20 {(2 - x)}^{\frac{3}{2}} - 6 {(2 - x)}^{\frac{5}{2}}}{15} + C$

Step 14

Reorder terms.

$- \frac{1}{15} (20 {(2 - x)}^{\frac{3}{2}} - 6 {(2 - x)}^{\frac{5}{2}}) + C$