Calculus Examples

$f (x) = 12 x^{3} + \frac{24}{x^{4}} - \frac{2}{\sqrt{x}}$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int 12 x^{3} + \frac{24}{x^{4}} - \frac{2}{\sqrt{x}} d x$

Step 3

Split the single integral into multiple integrals.

$\int 12 x^{3} d x + \int \frac{24}{x^{4}} d x + \int - \frac{2}{\sqrt{x}} d x$

Step 4

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int x^{3} d x + \int \frac{24}{x^{4}} d x + \int - \frac{2}{\sqrt{x}} d x$

Step 5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$12 (\frac{1}{4} x^{4} + C) + \int \frac{24}{x^{4}} d x + \int - \frac{2}{\sqrt{x}} d x$

Step 6

Since $24$ is constant with respect to $x$ , move $24$ out of the integral.

$12 (\frac{1}{4} x^{4} + C) + 24 \int \frac{1}{x^{4}} d x + \int - \frac{2}{\sqrt{x}} d x$

Step 7

Simplify the expression.

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Step 7.1

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

$12 (\frac{1}{4} x^{4} + C) + 24 \int {(x^{4})}^{- 1} d x + \int - \frac{2}{\sqrt{x}} d x$

Step 7.2

Simplify.

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Step 7.2.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$12 (\frac{x^{4}}{4} + C) + 24 \int {(x^{4})}^{- 1} d x + \int - \frac{2}{\sqrt{x}} d x$

Step 7.2.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 7.2.2.1

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

$12 (\frac{x^{4}}{4} + C) + 24 \int x^{4 \cdot - 1} d x + \int - \frac{2}{\sqrt{x}} d x$

Step 7.2.2.2

Multiply $4$ by $- 1$ .

$12 (\frac{x^{4}}{4} + C) + 24 \int x^{- 4} d x + \int - \frac{2}{\sqrt{x}} d x$

Step 8

By the Power Rule, the integral of $x^{- 4}$ with respect to $x$ is $- \frac{1}{3} x^{- 3}$ .

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{1}{3} x^{- 3} + C) + \int - \frac{2}{\sqrt{x}} d x$

Step 9

Simplify.

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Step 9.1

Combine $x^{- 3}$ and $\frac{1}{3}$ .

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{x^{- 3}}{3} + C) + \int - \frac{2}{\sqrt{x}} d x$

Step 9.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{1}{3 x^{3}} + C) + \int - \frac{2}{\sqrt{x}} d x$

Step 10

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

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{1}{3 x^{3}} + C) - \int \frac{2}{\sqrt{x}} d x$

Step 11

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{1}{3 x^{3}} + C) - (2 \int \frac{1}{\sqrt{x}} d x)$

Step 12

Simplify the expression.

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Step 12.1

Multiply $2$ by $- 1$ .

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{1}{3 x^{3}} + C) - 2 \int \frac{1}{\sqrt{x}} d x$

Step 12.2

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

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{1}{3 x^{3}} + C) - 2 \int \frac{1}{x^{\frac{1}{2}}} d x$

Step 12.3

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

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{1}{3 x^{3}} + C) - 2 \int {(x^{\frac{1}{2}})}^{- 1} d x$

Step 12.4

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

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Step 12.4.1

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

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{1}{3 x^{3}} + C) - 2 \int x^{\frac{1}{2} \cdot - 1} d x$

Step 12.4.2

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

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{1}{3 x^{3}} + C) - 2 \int x^{\frac{- 1}{2}} d x$

Step 12.4.3

Move the negative in front of the fraction.

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{1}{3 x^{3}} + C) - 2 \int x^{- \frac{1}{2}} d x$

Step 13

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

$12 (\frac{x^{4}}{4} + C) + 24 (- \frac{1}{3 x^{3}} + C) - 2 (2 x^{\frac{1}{2}} + C)$

Step 14

Simplify.

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Step 14.1

Simplify.

$3 x^{4} - \frac{8}{x^{3}} - 2 (2 x^{\frac{1}{2}}) + C$

Step 14.2

Multiply $2$ by $- 2$ .

$3 x^{4} - \frac{8}{x^{3}} - 4 x^{\frac{1}{2}} + C$

Step 15

The answer is the antiderivative of the function $f (x) = 12 x^{3} + \frac{24}{x^{4}} - \frac{2}{\sqrt{x}}$ .

$F (x) =$ $3 x^{4} - \frac{8}{x^{3}} - 4 x^{\frac{1}{2}} + C$