Calculus Examples

$\frac{1}{x^{\frac{3}{2}}}$

Step 1

Write $\frac{1}{x^{\frac{3}{2}}}$ as a function.

$f (x) = \frac{1}{x^{\frac{3}{2}}}$

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.

$F (x) = \int \frac{1}{x^{\frac{3}{2}}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

Multiply $3$ by $- 1$ .

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

Step 4.2.3

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

Move the negative in front of the fraction.

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

Step 7

The answer is the antiderivative of the function $f (x) = \frac{1}{x^{\frac{3}{2}}}$ .

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