Calculus Examples

$\int \frac{\ln (1 + \sqrt{x})}{\sqrt{x}} d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

$\int \frac{\ln (1 + x^{\frac{1}{2}})}{\sqrt{x}} d x$

Step 1.2

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

$\int \frac{\ln (1 + x^{\frac{1}{2}})}{x^{\frac{1}{2}}} d x$

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $1 + x^{\frac{1}{2}}$ .

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

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

$0 + \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [x^{\frac{1}{2}}]$ .

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Step 2.1.3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$0 + \frac{1}{2} x^{\frac{1}{2} - 1}$

Step 2.1.3.2

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

$0 + \frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 2.1.3.3

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

$0 + \frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 2.1.3.4

Combine the numerators over the common denominator.

$0 + \frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}$

Step 2.1.3.5

Simplify the numerator.

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Step 2.1.3.5.1

Multiply $- 1$ by $2$ .

$0 + \frac{1}{2} x^{\frac{1 - 2}{2}}$

Step 2.1.3.5.2

Subtract $2$ from $1$ .

$0 + \frac{1}{2} x^{\frac{- 1}{2}}$

Step 2.1.3.6

Move the negative in front of the fraction.

$0 + \frac{1}{2} x^{- \frac{1}{2}}$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0 + \frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}}$

Step 2.1.4.2

Combine terms.

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Step 2.1.4.2.1

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

$0 + \frac{1}{2 x^{\frac{1}{2}}}$

Step 2.1.4.2.2

Add $0$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{1}{2 x^{\frac{1}{2}}}$

Step 2.2

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

$\int 2 \ln (u) d u$

Step 3

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

$2 \int \ln (u) d u$

Step 4

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = \ln (u)$ and $dv = 1$ .

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

Step 5

Simplify.

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Step 5.1

Combine $u$ and $\frac{1}{u}$ .

$2 (\ln (u) u - \int \frac{u}{u} d u)$

Step 5.2

Cancel the common factor of $u$ .

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Step 5.2.1

Cancel the common factor.

$2 (\ln (u) u - \int \frac{u}{u} d u)$

Step 5.2.2

Rewrite the expression.

$2 (\ln (u) u - \int d u)$

Step 6

Apply the constant rule.

$2 (\ln (u) u - (u + C))$

Step 7

Simplify.

$2 (\ln (u) u - u) + C$

Step 8

Replace all occurrences of $u$ with $1 + x^{\frac{1}{2}}$ .

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Multiply $- 1$ by $1$ .

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

Step 9.3

Apply the distributive property.

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

Step 9.4

Simplify.

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Step 9.4.1

Multiply $2$ by $- 1$ .

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

Step 9.4.2

Multiply $- 1$ by $2$ .

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