Calculus Examples

$f (x) = \frac{2}{x^{3}} + 9 x^{5}$

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.

$F (x) = \int \frac{2}{x^{3}} + 9 x^{5} d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{2}{x^{3}} d x + \int 9 x^{5} d x$

Step 4

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

$2 \int \frac{1}{x^{3}} d x + \int 9 x^{5} d x$

Step 5

Apply basic rules of exponents.

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Step 5.1

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

$2 \int {(x^{3})}^{- 1} d x + \int 9 x^{5} d x$

Step 5.2

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

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Step 5.2.1

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

$2 \int x^{3 \cdot - 1} d x + \int 9 x^{5} d x$

Step 5.2.2

Multiply $3$ by $- 1$ .

$2 \int x^{- 3} d x + \int 9 x^{5} d x$

Step 6

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

$2 (- \frac{1}{2} x^{- 2} + C) + \int 9 x^{5} d x$

Step 7

Simplify.

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Step 7.1

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

$2 (- \frac{x^{- 2}}{2} + C) + \int 9 x^{5} d x$

Step 7.2

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

$2 (- \frac{1}{2 x^{2}} + C) + \int 9 x^{5} d x$

Step 8

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$2 (- \frac{1}{2 x^{2}} + C) + 9 \int x^{5} d x$

Step 9

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

$2 (- \frac{1}{2 x^{2}} + C) + 9 (\frac{1}{6} x^{6} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{- 1}{x^{2}} + 9 (\frac{1}{6} x^{6}) + C$

Step 10.2

Simplify.

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Step 10.2.1

Move the negative in front of the fraction.

$- \frac{1}{x^{2}} + 9 (\frac{1}{6} x^{6}) + C$

Step 10.2.2

Combine $\frac{1}{6}$ and $x^{6}$ .

$- \frac{1}{x^{2}} + 9 \frac{x^{6}}{6} + C$

Step 10.2.3

Combine $9$ and $\frac{x^{6}}{6}$ .

$- \frac{1}{x^{2}} + \frac{9 x^{6}}{6} + C$

Step 10.2.4

Cancel the common factor of $9$ and $6$ .

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Step 10.2.4.1

Factor $3$ out of $9 x^{6}$ .

$- \frac{1}{x^{2}} + \frac{3 (3 x^{6})}{6} + C$

Step 10.2.4.2

Cancel the common factors.

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Step 10.2.4.2.1

Factor $3$ out of $6$ .

$- \frac{1}{x^{2}} + \frac{3 (3 x^{6})}{3 (2)} + C$

Step 10.2.4.2.2

Cancel the common factor.

$- \frac{1}{x^{2}} + \frac{3 (3 x^{6})}{3 \cdot 2} + C$

Step 10.2.4.2.3

Rewrite the expression.

$- \frac{1}{x^{2}} + \frac{3 x^{6}}{2} + C$

Step 10.3

Reorder terms.

$- \frac{1}{x^{2}} + \frac{3}{2} x^{6} + C$

Step 11

The answer is the antiderivative of the function $f (x) = \frac{2}{x^{3}} + 9 x^{5}$ .

$F (x) =$ $- \frac{1}{x^{2}} + \frac{3}{2} x^{6} + C$