Calculus Examples

$\frac{x}{\sqrt{2 x + 1}}$

Step 1

Write $\frac{x}{\sqrt{2 x + 1}}$ as a function.

$f (x) = \frac{x}{\sqrt{2 x + 1}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{x}{\sqrt{2 x + 1}} d x$

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $2 x + 1$ .

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

Step 4.1.2

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 4.1.3

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

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Step 4.1.3.1

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 4.1.3.2

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 4.1.3.3

Multiply $2$ by $1$ .

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

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

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

$2 + 0$

Step 4.1.4.2

Add $2$ and $0$ .

$2$

Step 4.2

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 5

Simplify.

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Step 5.1

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 5.2

Combine.

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Rewrite the expression.

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

Step 5.5

Multiply $- 1$ by $2$ .

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

Step 5.6

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

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

Step 5.7

Cancel the common factor of $- 2$ and $2$ .

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Step 5.7.1

Factor $2$ out of $- 2$ .

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

Step 5.7.2

Cancel the common factors.

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Step 5.7.2.1

Factor $2$ out of $2$ .

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

Step 5.7.2.2

Cancel the common factor.

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

Step 5.7.2.3

Rewrite the expression.

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

Step 5.7.2.4

Divide $- 1$ by $1$ .

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

Step 5.8

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

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

Step 5.9

Multiply $2$ by $2$ .

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

Step 6

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 7

Apply basic rules of exponents.

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Step 7.1

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 7.2

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 7.3

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

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Step 7.3.1

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 7.3.2

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

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

Step 7.3.3

Move the negative in front of the fraction.

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

Step 8

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

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Step 8.1

Apply the distributive property.

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

Step 8.2

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 8.3

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 8.4

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 8.5

Combine the numerators over the common denominator.

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

Step 8.6

Subtract $1$ from $2$ .

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

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

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

Step 11

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

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

Step 12

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 13

Simplify.

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

Step 14

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

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

Step 15

Simplify.

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Step 15.1

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 15.2

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 15.3

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 15.4

Simplify the numerator.

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Step 15.4.1

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 15.4.1.1

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 15.4.1.2

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 15.4.1.3

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

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

Step 15.4.1.4

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

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

Step 15.4.2

Multiply $- 1$ by $3$ .

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

Step 15.4.3

Simplify each term.

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Step 15.4.3.1

Divide $2$ by $2$ .

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

Step 15.4.3.2

Simplify.

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

Step 15.4.4

Subtract $3$ from $1$ .

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

Step 15.4.5

Factor $2$ out of $2 x - 2$ .

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Step 15.4.5.1

Factor $2$ out of $2 x$ .

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

Step 15.4.5.2

Factor $2$ out of $- 2$ .

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

Step 15.4.5.3

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 15.4.6

Multiply $2$ by $2$ .

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

Step 15.5

Combine.

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

Step 15.6

Cancel the common factor.

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

Step 15.7

Rewrite the expression.

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

Step 15.8

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

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

Step 16

The answer is the antiderivative of the function $f (x) = \frac{x}{\sqrt{2 x + 1}}$ .

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