Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $5$ by $- 1$ .

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 8

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

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

Step 9

Simplify the expression.

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Step 9.1

Multiply $2$ by $- 1$ .

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

Step 9.2

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 9.3

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 9.3.1

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

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

Step 9.3.2

Multiply $2$ by $- 1$ .

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

Step 10

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

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

Step 11

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 12

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

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

Step 13

Multiply $6$ by $- 1$ .

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

Step 14

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$3 (- \frac{1}{4 x^{4}} + C) - 2 (- x^{- 1} + C) - 6 (\ln (| x |) + C)$

Step 15

Simplify.

$- \frac{3}{4 x^{4}} + \frac{2}{x} - 6 \ln (| x |) + C$