Calculus Examples

$\int (2 - \frac{1}{x^{5}} + \frac{7}{x^{3}}) d x$

Step 1

Remove parentheses.

$\int 2 - \frac{1}{x^{5}} + \frac{7}{x^{3}} d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 d x + \int - \frac{1}{x^{5}} d x + \int \frac{7}{x^{3}} d x$

Step 3

Apply the constant rule.

$2 x + C + \int - \frac{1}{x^{5}} d x + \int \frac{7}{x^{3}} d x$

Step 4

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

$2 x + C - \int \frac{1}{x^{5}} d x + \int \frac{7}{x^{3}} d x$

Step 5

Apply basic rules of exponents.

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Step 5.1

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

$2 x + C - \int {(x^{5})}^{- 1} d x + \int \frac{7}{x^{3}} d x$

Step 5.2

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

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Step 5.2.1

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

$2 x + C - \int x^{5 \cdot - 1} d x + \int \frac{7}{x^{3}} d x$

Step 5.2.2

Multiply $5$ by $- 1$ .

$2 x + C - \int x^{- 5} d x + \int \frac{7}{x^{3}} d x$

Step 6

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

$2 x + C - (- \frac{1}{4} x^{- 4} + C) + \int \frac{7}{x^{3}} d x$

Step 7

Simplify.

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Step 7.1

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

$2 x + C - (- \frac{x^{- 4}}{4} + C) + \int \frac{7}{x^{3}} d x$

Step 7.2

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

$2 x + C - (- \frac{1}{4 x^{4}} + C) + \int \frac{7}{x^{3}} d x$

Step 8

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$2 x + C - (- \frac{1}{4 x^{4}} + C) + 7 \int \frac{1}{x^{3}} d x$

Step 9

Apply basic rules of exponents.

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Step 9.1

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

$2 x + C - (- \frac{1}{4 x^{4}} + C) + 7 \int {(x^{3})}^{- 1} d x$

Step 9.2

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

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Step 9.2.1

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

$2 x + C - (- \frac{1}{4 x^{4}} + C) + 7 \int x^{3 \cdot - 1} d x$

Step 9.2.2

Multiply $3$ by $- 1$ .

$2 x + C - (- \frac{1}{4 x^{4}} + C) + 7 \int x^{- 3} d x$

Step 10

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

$2 x + C - (- \frac{1}{4 x^{4}} + C) + 7 (- \frac{1}{2} x^{- 2} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

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Step 11.1.1

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

$2 x + C - (- \frac{1}{4 x^{4}} + C) + 7 (- \frac{x^{- 2}}{2} + C)$

Step 11.1.2

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

$2 x + C - (- \frac{1}{4 x^{4}} + C) + 7 (- \frac{1}{2 x^{2}} + C)$

Step 11.2

Simplify.

$2 x + \frac{1}{4 x^{4}} + 7 (- \frac{1}{2 x^{2}}) + C$

Step 11.3

Simplify.

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Step 11.3.1

Multiply $- 1$ by $7$ .

$2 x + \frac{1}{4 x^{4}} - 7 \frac{1}{2 x^{2}} + C$

Step 11.3.2

Combine $- 7$ and $\frac{1}{2 x^{2}}$ .

$2 x + \frac{1}{4 x^{4}} + \frac{- 7}{2 x^{2}} + C$

Step 11.3.3

Move the negative in front of the fraction.

$2 x + \frac{1}{4 x^{4}} - \frac{7}{2 x^{2}} + C$