Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

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

$\int (x^{3} + \sqrt{x}) {(x^{5})}^{- 1} d x$

Step 3

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

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Step 3.1

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

$\int (x^{3} + \sqrt{x}) x^{5 \cdot - 1} d x$

Step 3.2

Multiply $5$ by $- 1$ .

$\int (x^{3} + \sqrt{x}) x^{- 5} d x$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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

Step 5.3

Subtract $5$ from $3$ .

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

Step 5.4

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

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

Step 5.5

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.6

Combine $- 5$ and $\frac{2}{2}$ .

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

Step 5.7

Combine the numerators over the common denominator.

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

Step 5.8

Simplify the numerator.

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Step 5.8.1

Multiply $- 5$ by $2$ .

$\int x^{- 2} + x^{\frac{1 - 10}{2}} d x$

Step 5.8.2

Subtract $10$ from $1$ .

$\int x^{- 2} + x^{\frac{- 9}{2}} d x$

Step 6

Move the negative in front of the fraction.

$\int x^{- 2} + x^{- \frac{9}{2}} d x$

Step 7

Split the single integral into multiple integrals.

$\int x^{- 2} d x + \int x^{- \frac{9}{2}} d x$

Step 8

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

$- x^{- 1} + C + \int x^{- \frac{9}{2}} d x$

Step 9

By the Power Rule, the integral of $x^{- \frac{9}{2}}$ with respect to $x$ is $- \frac{2}{7} x^{- \frac{7}{2}}$ .

$- x^{- 1} + C - \frac{2}{7} x^{- \frac{7}{2}} + C$

Step 10

Simplify.

$- \frac{1}{x} - \frac{2}{7} x^{- \frac{7}{2}} + C$