Calculus Examples

$g (x) = \frac{x^{2} + 3 x - 5}{x^{4}}$

Step 1

The function $G (x)$ can be found by finding the indefinite integral of the derivative $g (x)$ .

$G (x) = \int g (x) d x$

Step 2

Set up the integral to solve.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Multiply $4$ by $- 1$ .

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

Step 5

Expand $(x^{2} + 3 x - 5) x^{- 4}$ .

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.4

Subtract $4$ from $2$ .

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

Step 5.5

Raise $x$ to the power of $1$ .

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

Step 5.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.7

Subtract $4$ from $1$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

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

Step 12

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

$- x^{- 1} + C + 3 (- \frac{1}{2 x^{2}} + C) - 5 (- \frac{1}{3} x^{- 3} + C)$

Step 13

Simplify.

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Step 13.1

Simplify.

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Step 13.1.1

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

$- x^{- 1} + C + 3 (- \frac{1}{2 x^{2}} + C) - 5 (- \frac{x^{- 3}}{3} + C)$

Step 13.1.2

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

$- x^{- 1} + C + 3 (- \frac{1}{2 x^{2}} + C) - 5 (- \frac{1}{3 x^{3}} + C)$

Step 13.2

Simplify.

$- \frac{1}{x} - \frac{3}{2 x^{2}} - 5 (- \frac{1}{3 x^{3}}) + C$

Step 13.3

Simplify.

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Step 13.3.1

Multiply $- 1$ by $- 5$ .

$- \frac{1}{x} - \frac{3}{2 x^{2}} + 5 \frac{1}{3 x^{3}} + C$

Step 13.3.2

Combine $5$ and $\frac{1}{3 x^{3}}$ .

$- \frac{1}{x} - \frac{3}{2 x^{2}} + \frac{5}{3 x^{3}} + C$

Step 14

The answer is the antiderivative of the function $g (x) = \frac{x^{2} + 3 x - 5}{x^{4}}$ .

$G (x) =$ $- \frac{1}{x} - \frac{3}{2 x^{2}} + \frac{5}{3 x^{3}} + C$