Calculus Examples

$\int \frac{4}{x^{3}} - 5 x \sqrt{x} d x$

Step 1

Simplify.

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Step 1.1

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

$\int \frac{4}{x^{3}} - 5 x \cdot x^{\frac{1}{2}} d x$

Step 1.2

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

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Step 1.2.1

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

$\int \frac{4}{x^{3}} - 5 (x^{\frac{1}{2}} x) d x$

Step 1.2.2

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

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Step 1.2.2.1

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

$\int \frac{4}{x^{3}} - 5 (x^{\frac{1}{2}} x^{1}) d x$

Step 1.2.2.2

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

$\int \frac{4}{x^{3}} - 5 x^{\frac{1}{2} + 1} d x$

Step 1.2.3

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

$\int \frac{4}{x^{3}} - 5 x^{\frac{1}{2} + \frac{2}{2}} d x$

Step 1.2.4

Combine the numerators over the common denominator.

$\int \frac{4}{x^{3}} - 5 x^{\frac{1 + 2}{2}} d x$

Step 1.2.5

Add $1$ and $2$ .

$\int \frac{4}{x^{3}} - 5 x^{\frac{3}{2}} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{4}{x^{3}} d x + \int - 5 x^{\frac{3}{2}} d x$

Step 3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int \frac{1}{x^{3}} d x + \int - 5 x^{\frac{3}{2}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$4 \int {(x^{3})}^{- 1} d x + \int - 5 x^{\frac{3}{2}} d x$

Step 4.2

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

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Step 4.2.1

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

$4 \int x^{3 \cdot - 1} d x + \int - 5 x^{\frac{3}{2}} d x$

Step 4.2.2

Multiply $3$ by $- 1$ .

$4 \int x^{- 3} d x + \int - 5 x^{\frac{3}{2}} d x$

Step 5

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

$4 (- \frac{1}{2} x^{- 2} + C) + \int - 5 x^{\frac{3}{2}} d x$

Step 6

Simplify.

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Step 6.1

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

$4 (- \frac{x^{- 2}}{2} + C) + \int - 5 x^{\frac{3}{2}} d x$

Step 6.2

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

$4 (- \frac{1}{2 x^{2}} + C) + \int - 5 x^{\frac{3}{2}} d x$

Step 7

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$4 (- \frac{1}{2 x^{2}} + C) - 5 \int x^{\frac{3}{2}} d x$

Step 8

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

$4 (- \frac{1}{2 x^{2}} + C) - 5 (\frac{2}{5} x^{\frac{5}{2}} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Simplify.

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Step 9.2.1

Move the negative in front of the fraction.

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

Step 9.2.2

Combine $\frac{2}{5}$ and $x^{\frac{5}{2}}$ .

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

Step 9.2.3

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

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

Step 9.2.4

Multiply $2$ by $- 5$ .

$- \frac{2}{x^{2}} + \frac{- 10 x^{\frac{5}{2}}}{5} + C$

Step 9.2.5

Factor $5$ out of $- 10 x^{\frac{5}{2}}$ .

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

Step 9.2.6

Cancel the common factors.

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Step 9.2.6.1

Factor $5$ out of $5$ .

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

Step 9.2.6.2

Cancel the common factor.

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

Step 9.2.6.3

Rewrite the expression.

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

Step 9.2.6.4

Divide $- 2 x^{\frac{5}{2}}$ by $1$ .

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