Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.2.3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 6

By the Power Rule, the integral of $x^{\frac{4}{5}}$ with respect to $x$ is $\frac{5}{9} x^{\frac{9}{5}}$ .

$- 2 x^{- \frac{1}{2}} + C - (\frac{5}{9} x^{\frac{9}{5}} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$- 2 \frac{1}{x^{\frac{1}{2}}} - (\frac{5}{9} x^{\frac{9}{5}}) + C$

Step 7.2

Simplify.

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Step 7.2.1

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

$\frac{- 2}{x^{\frac{1}{2}}} - (\frac{5}{9} x^{\frac{9}{5}}) + C$

Step 7.2.2

Move the negative in front of the fraction.

$- \frac{2}{x^{\frac{1}{2}}} - (\frac{5}{9} x^{\frac{9}{5}}) + C$

Step 7.2.3

Combine $\frac{5}{9}$ and $x^{\frac{9}{5}}$ .

$- \frac{2}{x^{\frac{1}{2}}} - \frac{5 x^{\frac{9}{5}}}{9} + C$

Step 8

Reorder terms.

$- \frac{2}{x^{\frac{1}{2}}} - \frac{5}{9} x^{\frac{9}{5}} + C$