Calculus Examples

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

Step 1

Write $- \frac{3}{x^{2}} + \frac{4}{x^{3}}$ as a function.

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the expression.

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Step 7.1

Multiply $3$ by $- 1$ .

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

Step 7.2

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

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

Multiply $2$ by $- 1$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Apply basic rules of exponents.

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Step 10.1

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

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

Step 10.2

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

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Step 10.2.1

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

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

Step 10.2.2

Multiply $3$ by $- 1$ .

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

Simplify.

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Step 12.1.1

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

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

Step 12.1.2

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

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

Step 12.2

Simplify.

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

Step 12.3

Simplify.

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Step 12.3.1

Multiply $- 1$ by $- 3$ .

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

Step 12.3.2

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

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

Step 12.3.3

Multiply $- 1$ by $4$ .

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

Step 12.3.4

Combine $- 4$ and $\frac{1}{2 x^{2}}$ .

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

Step 12.3.5

Cancel the common factor of $- 4$ and $2$ .

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Step 12.3.5.1

Factor $2$ out of $- 4$ .

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

Step 12.3.5.2

Cancel the common factors.

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Step 12.3.5.2.1

Factor $2$ out of $2 x^{2}$ .

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

Step 12.3.5.2.2

Cancel the common factor.

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

Step 12.3.5.2.3

Rewrite the expression.

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

Step 12.3.6

Move the negative in front of the fraction.

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

Step 13

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

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