Calculus Examples

$f (x) = \frac{9}{x^{10}} + \frac{8}{x^{9}}$

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.

$F (x) = \int \frac{9}{x^{10}} + \frac{8}{x^{9}} d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{9}{x^{10}} d x + \int \frac{8}{x^{9}} d x$

Step 4

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

$9 \int \frac{1}{x^{10}} d x + \int \frac{8}{x^{9}} d x$

Step 5

Apply basic rules of exponents.

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Step 5.1

Move $x^{10}$ out of the denominator by raising it to the $- 1$ power.

$9 \int {(x^{10})}^{- 1} d x + \int \frac{8}{x^{9}} d x$

Step 5.2

Multiply the exponents in ${(x^{10})}^{- 1}$ .

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Step 5.2.1

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

$9 \int x^{10 \cdot - 1} d x + \int \frac{8}{x^{9}} d x$

Step 5.2.2

Multiply $10$ by $- 1$ .

$9 \int x^{- 10} d x + \int \frac{8}{x^{9}} d x$

Step 6

By the Power Rule, the integral of $x^{- 10}$ with respect to $x$ is $- \frac{1}{9} x^{- 9}$ .

$9 (- \frac{1}{9} x^{- 9} + C) + \int \frac{8}{x^{9}} d x$

Step 7

Simplify.

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Step 7.1

Combine $x^{- 9}$ and $\frac{1}{9}$ .

$9 (- \frac{x^{- 9}}{9} + C) + \int \frac{8}{x^{9}} d x$

Step 7.2

Move $x^{- 9}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$9 (- \frac{1}{9 x^{9}} + C) + \int \frac{8}{x^{9}} d x$

Step 8

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$9 (- \frac{1}{9 x^{9}} + C) + 8 \int \frac{1}{x^{9}} d x$

Step 9

Apply basic rules of exponents.

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Step 9.1

Move $x^{9}$ out of the denominator by raising it to the $- 1$ power.

$9 (- \frac{1}{9 x^{9}} + C) + 8 \int {(x^{9})}^{- 1} d x$

Step 9.2

Multiply the exponents in ${(x^{9})}^{- 1}$ .

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Step 9.2.1

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

$9 (- \frac{1}{9 x^{9}} + C) + 8 \int x^{9 \cdot - 1} d x$

Step 9.2.2

Multiply $9$ by $- 1$ .

$9 (- \frac{1}{9 x^{9}} + C) + 8 \int x^{- 9} d x$

Step 10

By the Power Rule, the integral of $x^{- 9}$ with respect to $x$ is $- \frac{1}{8} x^{- 8}$ .

$9 (- \frac{1}{9 x^{9}} + C) + 8 (- \frac{1}{8} x^{- 8} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

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Step 11.1.1

Combine $x^{- 8}$ and $\frac{1}{8}$ .

$9 (- \frac{1}{9 x^{9}} + C) + 8 (- \frac{x^{- 8}}{8} + C)$

Step 11.1.2

Move $x^{- 8}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$9 (- \frac{1}{9 x^{9}} + C) + 8 (- \frac{1}{8 x^{8}} + C)$

Step 11.2

Simplify.

$\frac{- 1}{x^{9}} + 8 (- \frac{1}{8 x^{8}}) + C$

Step 11.3

Simplify.

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Step 11.3.1

Move the negative in front of the fraction.

$- \frac{1}{x^{9}} + 8 (- \frac{1}{8 x^{8}}) + C$

Step 11.3.2

Multiply $- 1$ by $8$ .

$- \frac{1}{x^{9}} - 8 \frac{1}{8 x^{8}} + C$

Step 11.3.3

Combine $- 8$ and $\frac{1}{8 x^{8}}$ .

$- \frac{1}{x^{9}} + \frac{- 8}{8 x^{8}} + C$

Step 11.3.4

Cancel the common factor of $- 8$ and $8$ .

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Step 11.3.4.1

Factor $8$ out of $- 8$ .

$- \frac{1}{x^{9}} + \frac{8 \cdot - 1}{8 x^{8}} + C$

Step 11.3.4.2

Cancel the common factors.

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Step 11.3.4.2.1

Factor $8$ out of $8 x^{8}$ .

$- \frac{1}{x^{9}} + \frac{8 \cdot - 1}{8 (x^{8})} + C$

Step 11.3.4.2.2

Cancel the common factor.

$- \frac{1}{x^{9}} + \frac{8 \cdot - 1}{8 x^{8}} + C$

Step 11.3.4.2.3

Rewrite the expression.

$- \frac{1}{x^{9}} + \frac{- 1}{x^{8}} + C$

Step 11.3.5

Move the negative in front of the fraction.

$- \frac{1}{x^{9}} - \frac{1}{x^{8}} + C$

Step 12

The answer is the antiderivative of the function $f (x) = \frac{9}{x^{10}} + \frac{8}{x^{9}}$ .

$F (x) =$ $- \frac{1}{x^{9}} - \frac{1}{x^{8}} + C$