Calculus Examples

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

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify the expression.

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Step 9.1

Multiply $2$ by $- 1$ .

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

Step 9.2

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

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

Step 9.3

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

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Step 9.3.1

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

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

Step 9.3.2

Multiply $3$ by $- 1$ .

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify.

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Step 11.1.1

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

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

Step 11.1.2

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

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

Step 11.2

Simplify.

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

Step 11.3

Simplify.

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Step 11.3.1

Multiply $- 1$ by $3$ .

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

Step 11.3.2

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

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

Step 11.3.3

Move the negative in front of the fraction.

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

Step 11.3.4

Multiply $- 1$ by $- 2$ .

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

Step 11.3.5

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

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

Step 11.3.6

Cancel the common factor of $2$ .

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Step 11.3.6.1

Cancel the common factor.

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

Step 11.3.6.2

Rewrite the expression.

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

Step 12

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

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