Calculus Examples

$f (x) = \frac{\sqrt{2 x}}{x} + \frac{3}{x^{5}}$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int \frac{\sqrt{2 x}}{x} + \frac{3}{x^{5}} d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{\sqrt{2 x}}{x} d x + \int \frac{3}{x^{5}} d x$

Step 4

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

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Step 4.1.1

Differentiate $2 x$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

$2 \cdot 1$

Step 4.1.4

Multiply $2$ by $1$ .

$2$

Step 4.2

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

$\int \frac{\sqrt{u}}{\frac{u}{2}} \cdot \frac{1}{2} d u + \int \frac{3}{x^{5}} d x$

Step 5

Simplify the expression.

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Step 5.1

Simplify.

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Step 5.1.1

Multiply by the reciprocal of the fraction to divide by $\frac{u}{2}$ .

$\int \sqrt{u} \frac{2}{u} \frac{1}{2} d u + \int \frac{3}{x^{5}} d x$

Step 5.1.2

Combine $\sqrt{u}$ and $\frac{2}{u}$ .

$\int \frac{\sqrt{u} \cdot 2}{u} \cdot \frac{1}{2} d u + \int \frac{3}{x^{5}} d x$

Step 5.1.3

Move $2$ to the left of $\sqrt{u}$ .

$\int \frac{2 \cdot \sqrt{u}}{u} \cdot \frac{1}{2} d u + \int \frac{3}{x^{5}} d x$

Step 5.1.4

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

$\int \frac{2 \sqrt{u}}{u \cdot 2} d u + \int \frac{3}{x^{5}} d x$

Step 5.1.5

Move $2$ to the left of $u$ .

$\int \frac{2 \sqrt{u}}{2 \cdot u} d u + \int \frac{3}{x^{5}} d x$

Step 5.1.6

Cancel the common factor of $2$ .

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Step 5.1.6.1

Cancel the common factor.

$\int \frac{2 \sqrt{u}}{2 u} d u + \int \frac{3}{x^{5}} d x$

Step 5.1.6.2

Rewrite the expression.

$\int \frac{\sqrt{u}}{u} d u + \int \frac{3}{x^{5}} d x$

Step 5.2

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

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

Step 5.3

Simplify.

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Step 5.3.1

Move $u^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 5.3.2

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

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Step 5.3.2.1

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

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Step 5.3.2.1.1

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

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

Step 5.3.2.1.2

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

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

Step 5.3.2.2

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

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

Step 5.3.2.3

Combine the numerators over the common denominator.

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

Step 5.3.2.4

Subtract $1$ from $2$ .

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

Step 5.4

Apply basic rules of exponents.

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Step 5.4.1

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

$\int {(u^{\frac{1}{2}})}^{- 1} d u + \int \frac{3}{x^{5}} d x$

Step 5.4.2

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

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Step 5.4.2.1

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

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

Step 5.4.2.2

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

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

Step 5.4.2.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

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

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

Step 8

Apply basic rules of exponents.

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Step 8.1

Move $x^{5}$ out of the denominator by raising it to the $- 1$ power.

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

Step 8.2

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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Step 8.2.1

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

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

Step 8.2.2

Multiply $5$ by $- 1$ .

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

Step 9

By the Power Rule, the integral of $x^{- 5}$ with respect to $x$ is $- \frac{1}{4} x^{- 4}$ .

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

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $x^{- 4}$ and $\frac{1}{4}$ .

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

Step 10.1.2

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 10.2

Simplify.

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

Step 10.3

Move the negative in front of the fraction.

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

Step 11

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

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

Step 12

The answer is the antiderivative of the function $f (x) = \frac{\sqrt{2 x}}{x} + \frac{3}{x^{5}}$ .

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