Calculus Examples

$\int (\frac{15}{x^{4}} + \frac{8}{x^{5}}) d x$

Step 1

Remove parentheses.

$\int \frac{15}{x^{4}} + \frac{8}{x^{5}} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{15}{x^{4}} d x + \int \frac{8}{x^{5}} d x$

Step 3

Since $15$ is constant with respect to $x$ , move $15$ out of the integral.

$15 \int \frac{1}{x^{4}} d x + \int \frac{8}{x^{5}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$15 \int {(x^{4})}^{- 1} d x + \int \frac{8}{x^{5}} d x$

Step 4.2

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

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Step 4.2.1

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

$15 \int x^{4 \cdot - 1} d x + \int \frac{8}{x^{5}} d x$

Step 4.2.2

Multiply $4$ by $- 1$ .

$15 \int x^{- 4} d x + \int \frac{8}{x^{5}} d x$

Step 5

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

$15 (- \frac{1}{3} x^{- 3} + C) + \int \frac{8}{x^{5}} d x$

Step 6

Simplify.

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Step 6.1

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

$15 (- \frac{x^{- 3}}{3} + C) + \int \frac{8}{x^{5}} d x$

Step 6.2

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

$15 (- \frac{1}{3 x^{3}} + C) + \int \frac{8}{x^{5}} d x$

Step 7

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

$15 (- \frac{1}{3 x^{3}} + C) + 8 \int \frac{1}{x^{5}} d x$

Step 8

Apply basic rules of exponents.

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Step 8.1

Move $x^{5}$ out of the denominator by raising it to the $- 1$ power.

$15 (- \frac{1}{3 x^{3}} + C) + 8 \int {(x^{5})}^{- 1} d x$

Step 8.2

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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Step 8.2.1

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

$15 (- \frac{1}{3 x^{3}} + C) + 8 \int x^{5 \cdot - 1} d x$

Step 8.2.2

Multiply $5$ by $- 1$ .

$15 (- \frac{1}{3 x^{3}} + C) + 8 \int x^{- 5} d x$

Step 9

By the Power Rule, the integral of $x^{- 5}$ with respect to $x$ is $- \frac{1}{4} x^{- 4}$ .

$15 (- \frac{1}{3 x^{3}} + C) + 8 (- \frac{1}{4} x^{- 4} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

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

$15 (- \frac{1}{3 x^{3}} + C) + 8 (- \frac{x^{- 4}}{4} + C)$

Step 10.1.2

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$15 (- \frac{1}{3 x^{3}} + C) + 8 (- \frac{1}{4 x^{4}} + C)$

Step 10.2

Simplify.

$\frac{- 5}{x^{3}} + 8 (- \frac{1}{4 x^{4}}) + C$

Step 10.3

Simplify.

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Step 10.3.1

Move the negative in front of the fraction.

$- \frac{5}{x^{3}} + 8 (- \frac{1}{4 x^{4}}) + C$

Step 10.3.2

Multiply $- 1$ by $8$ .

$- \frac{5}{x^{3}} - 8 \frac{1}{4 x^{4}} + C$

Step 10.3.3

Combine $- 8$ and $\frac{1}{4 x^{4}}$ .

$- \frac{5}{x^{3}} + \frac{- 8}{4 x^{4}} + C$

Step 10.3.4

Cancel the common factor of $- 8$ and $4$ .

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Step 10.3.4.1

Factor $4$ out of $- 8$ .

$- \frac{5}{x^{3}} + \frac{4 \cdot - 2}{4 x^{4}} + C$

Step 10.3.4.2

Cancel the common factors.

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Step 10.3.4.2.1

Factor $4$ out of $4 x^{4}$ .

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

Step 10.3.4.2.2

Cancel the common factor.

$- \frac{5}{x^{3}} + \frac{4 \cdot - 2}{4 x^{4}} + C$

Step 10.3.4.2.3

Rewrite the expression.

$- \frac{5}{x^{3}} + \frac{- 2}{x^{4}} + C$

Step 10.3.5

Move the negative in front of the fraction.

$- \frac{5}{x^{3}} - \frac{2}{x^{4}} + C$