Calculus Examples

$f (x) = \frac{5 - 4 x^{3} + 2 x^{6}}{x^{6}}$

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.

$F (x) = \int \frac{5 - 4 x^{3} + 2 x^{6}}{x^{6}} d x$

Step 3

Move $x^{6}$ out of the denominator by raising it to the $- 1$ power.

$\int (5 - 4 x^{3} + 2 x^{6}) {(x^{6})}^{- 1} d x$

Step 4

Multiply the exponents in ${(x^{6})}^{- 1}$ .

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Step 4.1

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

$\int (5 - 4 x^{3} + 2 x^{6}) x^{6 \cdot - 1} d x$

Step 4.2

Multiply $6$ by $- 1$ .

$\int (5 - 4 x^{3} + 2 x^{6}) x^{- 6} d x$

Step 5

Expand $(5 - 4 x^{3} + 2 x^{6}) x^{- 6}$ .

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Step 5.1

Apply the distributive property.

$\int (5 - 4 x^{3}) x^{- 6} + 2 x^{6} x^{- 6} d x$

Step 5.2

Apply the distributive property.

$\int 5 x^{- 6} - 4 x^{3} x^{- 6} + 2 x^{6} x^{- 6} d x$

Step 5.3

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

$\int 5 x^{- 6} - 4 x^{3 - 6} + 2 x^{6} x^{- 6} d x$

Step 5.4

Subtract $6$ from $3$ .

$\int 5 x^{- 6} - 4 x^{- 3} + 2 x^{6} x^{- 6} d x$

Step 5.5

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

$\int 5 x^{- 6} - 4 x^{- 3} + 2 x^{6 - 6} d x$

Step 5.6

Subtract $6$ from $6$ .

$\int 5 x^{- 6} - 4 x^{- 3} + 2 x^{0} d x$

Step 5.7

Anything raised to $0$ is $1$ .

$\int 5 x^{- 6} - 4 x^{- 3} + 2 \cdot 1 d x$

Step 5.8

Multiply $2$ by $1$ .

$\int 5 x^{- 6} - 4 x^{- 3} + 2 d x$

Step 5.9

Reorder $5 x^{- 6}$ and $- 4 x^{- 3}$ .

$\int - 4 x^{- 3} + 5 x^{- 6} + 2 d x$

Step 6

Split the single integral into multiple integrals.

$\int - 4 x^{- 3} d x + \int 5 x^{- 6} d x + \int 2 d x$

Step 7

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

$- 4 \int x^{- 3} d x + \int 5 x^{- 6} d x + \int 2 d x$

Step 8

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

$- 4 (- \frac{1}{2} x^{- 2} + C) + \int 5 x^{- 6} d x + \int 2 d x$

Step 9

Simplify.

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Step 9.1

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

$- 4 (- \frac{x^{- 2}}{2} + C) + \int 5 x^{- 6} d x + \int 2 d x$

Step 9.2

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

$- 4 (- \frac{1}{2 x^{2}} + C) + \int 5 x^{- 6} d x + \int 2 d x$

Step 10

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

$- 4 (- \frac{1}{2 x^{2}} + C) + 5 \int x^{- 6} d x + \int 2 d x$

Step 11

By the Power Rule, the integral of $x^{- 6}$ with respect to $x$ is $- \frac{1}{5} x^{- 5}$ .

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

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 13

Apply the constant rule.

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

Step 14

Simplify.

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

Step 15

The answer is the antiderivative of the function $f (x) = \frac{5 - 4 x^{3} + 2 x^{6}}{x^{6}}$ .

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