Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.2

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

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

Add $2$ and $1$ .

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

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

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

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Step 4.4.1.1

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

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

Step 4.4.1.2

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

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

Step 4.4.2

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

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

Step 4.4.3

Combine the numerators over the common denominator.

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

Step 4.4.4

Add $2$ and $1$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

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

Step 7

Apply basic rules of exponents.

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Step 7.1

Move $x^{\frac{3}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 7.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{- 1}$ .

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Step 7.2.1

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

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

Step 7.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 7.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 7.2.2.2

Multiply $3$ by $- 1$ .

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

Step 7.2.3

Move the negative in front of the fraction.

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

Step 8

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 2 x^{- \frac{1}{2}} + C$

Step 9

Simplify.

$\frac{2}{5} x^{\frac{5}{2}} - 2 x^{- \frac{1}{2}} + C$

Step 10

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

$F (x) =$ $\frac{2}{5} x^{\frac{5}{2}} - 2 x^{- \frac{1}{2}} + C$