Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $3$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Apply basic rules of exponents.

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $2$ by $- 1$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

Multiply $\frac{1}{x^{2}}$ by $\frac{1}{2}$ .

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

Step 8.2.2

Move $2$ to the left of $x^{2}$ .

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

Step 8.2.3

Multiply $- 1$ by $- 1$ .

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

Step 8.2.4

Multiply $x^{- 1}$ by $1$ .

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