Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Factor $\sqrt{x}$ out of $x \sqrt{x} + \sqrt{x}$ .

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Step 4.1

Factor $\sqrt{x}$ out of $x \sqrt{x}$ .

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

Step 4.2

Multiply by $1$ .

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

Step 4.3

Factor $\sqrt{x}$ out of $\sqrt{x} x + \sqrt{x} \cdot 1$ .

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

Step 5

Simplify the expression.

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Step 5.1

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

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

Step 5.2

Simplify.

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Step 5.2.1

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

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Step 5.2.1.1

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

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

Step 5.2.1.2

Multiply $2$ by $- 1$ .

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

Step 5.2.2

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

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

Step 5.2.3

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

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Step 5.2.3.1

Move $x^{- 2}$ .

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

Step 5.2.3.2

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

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

Step 5.2.3.3

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

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

Step 5.2.3.4

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

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

Step 5.2.3.5

Combine the numerators over the common denominator.

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

Step 5.2.3.6

Simplify the numerator.

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Step 5.2.3.6.1

Multiply $- 2$ by $2$ .

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

Step 5.2.3.6.2

Add $- 4$ and $1$ .

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

Step 5.2.3.7

Move the negative in front of the fraction.

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

Step 6

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

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Step 6.1

Apply the distributive property.

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

Step 6.2

Reorder $x^{- \frac{3}{2}}$ and $1$ .

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

Combine the numerators over the common denominator.

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

Step 6.7

Add $- 3$ and $2$ .

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

Step 6.8

Multiply $x^{- \frac{3}{2}}$ by $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Simplify.

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Step 11.2.1

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

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

Step 11.2.2

Move the negative in front of the fraction.

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

Step 12

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

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