Calculus Examples

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

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 $d u$ .

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

Step 2

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

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Move parentheses.

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

Step 3.5

Multiply $- 1$ by $- 1$ .

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

Step 3.6

Multiply $u$ by $1$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Combine the numerators over the common denominator.

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

Step 3.11

Add $2$ and $1$ .

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

Step 3.12

Multiply $- 1$ by $2$ .

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

Step 3.13

Subtract $1 u^{\frac{1}{2}}$ from $- 2 u^{\frac{1}{2}}$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

Since $- 3$ is constant with respect to $u$ , move $- 3$ out of the integral.

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

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

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

Step 8.2.2

Multiply $- 3$ by $2$ .

$\frac{2}{5} u^{\frac{5}{2}} + \frac{- 6}{3} u^{\frac{3}{2}} + C$

Step 8.2.3

Cancel the common factor of $- 6$ and $3$ .

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Step 8.2.3.1

Factor $3$ out of $- 6$ .

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

Step 8.2.3.2

Cancel the common factors.

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Step 8.2.3.2.1

Factor $3$ out of $3$ .

$\frac{2}{5} u^{\frac{5}{2}} + \frac{3 \cdot - 2}{3 (1)} u^{\frac{3}{2}} + C$

Step 8.2.3.2.2

Cancel the common factor.

$\frac{2}{5} u^{\frac{5}{2}} + \frac{3 \cdot - 2}{3 \cdot 1} u^{\frac{3}{2}} + C$

Step 8.2.3.2.3

Rewrite the expression.

$\frac{2}{5} u^{\frac{5}{2}} + \frac{- 2}{1} u^{\frac{3}{2}} + C$

Step 8.2.3.2.4

Divide $- 2$ by $1$ .

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

Step 9

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

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