Calculus Examples

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

Step 1

Let $u = 2 x - 1$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x - 1$ .

$\frac{d}{d x} [2 x - 1]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

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 $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [- 1]$

Step 1.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 + 0$

Step 1.1.4.2

Add $2$ and $0$ .

$2$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

Multiply $\frac{\frac{u}{2} + \frac{1}{2}}{\sqrt{u}}$ by $\frac{2}{2}$ .

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

Step 2.2

Combine.

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

Step 2.3

Apply the distributive property.

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

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

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

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

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

Step 2.6

Multiply $\frac{u + 1}{2 \sqrt{u}}$ by $\frac{1}{2}$ .

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

Step 2.7

Multiply $2$ by $2$ .

$\int \frac{u + 1}{4 \sqrt{u}} d u$

Step 3

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

$\frac{1}{4} \int \frac{u + 1}{\sqrt{u}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{4} \int (u + 1) {(u^{\frac{1}{2}})}^{- 1} d u$

Step 4.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{4} \int (u + 1) u^{\frac{1}{2} \cdot - 1} d u$

Step 4.3.2

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

$\frac{1}{4} \int (u + 1) u^{\frac{- 1}{2}} d u$

Step 4.3.3

Move the negative in front of the fraction.

$\frac{1}{4} \int (u + 1) u^{- \frac{1}{2}} d u$

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Combine the numerators over the common denominator.

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

Step 5.6

Subtract $1$ from $2$ .

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

Step 5.7

Multiply $u^{- \frac{1}{2}}$ by $1$ .

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

Step 6

Split the single integral into multiple integrals.

$\frac{1}{4} (\int u^{\frac{1}{2}} d u + \int u^{- \frac{1}{2}} d u)$

Step 7

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

$\frac{1}{4} (\frac{2}{3} u^{\frac{3}{2}} + C + \int u^{- \frac{1}{2}} d u)$

Step 8

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

$\frac{1}{4} (\frac{2}{3} u^{\frac{3}{2}} + C + 2 u^{\frac{1}{2}} + C)$

Step 9

Simplify.

$\frac{1}{4} (\frac{2}{3} u^{\frac{3}{2}} + 2 u^{\frac{1}{2}}) + C$

Step 10

Replace all occurrences of $u$ with $2 x - 1$ .

$\frac{1}{4} (\frac{2}{3} {(2 x - 1)}^{\frac{3}{2}} + 2 {(2 x - 1)}^{\frac{1}{2}}) + C$

Step 11

Simplify.

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Step 11.1

Combine $\frac{2}{3}$ and ${(2 x - 1)}^{\frac{3}{2}}$ .

$\frac{1}{4} (\frac{2 {(2 x - 1)}^{\frac{3}{2}}}{3} + 2 {(2 x - 1)}^{\frac{1}{2}}) + C$

Step 11.2

To write $2 {(2 x - 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{4} (\frac{2 {(2 x - 1)}^{\frac{3}{2}}}{3} + 2 {(2 x - 1)}^{\frac{1}{2}} \cdot \frac{3}{3}) + C$

Step 11.3

Combine $2 {(2 x - 1)}^{\frac{1}{2}}$ and $\frac{3}{3}$ .

$\frac{1}{4} (\frac{2 {(2 x - 1)}^{\frac{3}{2}}}{3} + \frac{2 {(2 x - 1)}^{\frac{1}{2}} \cdot 3}{3}) + C$

Step 11.4

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{3}{2}} + 2 {(2 x - 1)}^{\frac{1}{2}} \cdot 3}{3} + C$

Step 11.5

Simplify the numerator.

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Step 11.5.1

Factor $2 {(2 x - 1)}^{\frac{1}{2}}$ out of $2 {(2 x - 1)}^{\frac{3}{2}} + 2 {(2 x - 1)}^{\frac{1}{2}} \cdot 3$ .

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Step 11.5.1.1

Move ${(2 x - 1)}^{\frac{1}{2}}$ .

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{3}{2}} + 2 \cdot 3 {(2 x - 1)}^{\frac{1}{2}}}{3} + C$

Step 11.5.1.2

Factor $2 {(2 x - 1)}^{\frac{1}{2}}$ out of $2 {(2 x - 1)}^{\frac{3}{2}}$ .

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{1}{2}} ({(2 x - 1)}^{\frac{2}{2}}) + 2 \cdot 3 {(2 x - 1)}^{\frac{1}{2}}}{3} + C$

Step 11.5.1.3

Factor $2 {(2 x - 1)}^{\frac{1}{2}}$ out of $2 \cdot 3 {(2 x - 1)}^{\frac{1}{2}}$ .

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{1}{2}} ({(2 x - 1)}^{\frac{2}{2}}) + 2 {(2 x - 1)}^{\frac{1}{2}} (3)}{3} + C$

Step 11.5.1.4

Factor $2 {(2 x - 1)}^{\frac{1}{2}}$ out of $2 {(2 x - 1)}^{\frac{1}{2}} ({(2 x - 1)}^{\frac{2}{2}}) + 2 {(2 x - 1)}^{\frac{1}{2}} (3)$ .

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{1}{2}} ({(2 x - 1)}^{\frac{2}{2}} + 3)}{3} + C$

Step 11.5.2

Divide $2$ by $2$ .

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{1}{2}} ({(2 x - 1)}^{1} + 3)}{3} + C$

Step 11.5.3

Simplify.

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{1}{2}} (2 x - 1 + 3)}{3} + C$

Step 11.5.4

Add $- 1$ and $3$ .

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{1}{2}} (2 x + 2)}{3} + C$

Step 11.5.5

Factor $2$ out of $2 x + 2$ .

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Step 11.5.5.1

Factor $2$ out of $2 x$ .

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{1}{2}} (2 (x) + 2)}{3} + C$

Step 11.5.5.2

Factor $2$ out of $2$ .

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{1}{2}} (2 (x) + 2 (1))}{3} + C$

Step 11.5.5.3

Factor $2$ out of $2 (x) + 2 (1)$ .

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{1}{2}} (2 (x + 1))}{3} + C$

$\frac{1}{4} \cdot \frac{2 {(2 x - 1)}^{\frac{1}{2}} \cdot 2 (x + 1)}{3} + C$

Step 11.5.6

Multiply $2$ by $2$ .

$\frac{1}{4} \cdot \frac{4 {(2 x - 1)}^{\frac{1}{2}} (x + 1)}{3} + C$

Step 11.6

Combine.

$\frac{1 (4 {(2 x - 1)}^{\frac{1}{2}} (x + 1))}{4 \cdot 3} + C$

Step 11.7

Cancel the common factor.

$\frac{1 (4 {(2 x - 1)}^{\frac{1}{2}} (x + 1))}{4 \cdot 3} + C$

Step 11.8

Rewrite the expression.

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

Step 11.9

Multiply ${(2 x - 1)}^{\frac{1}{2}}$ by $1$ .

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