Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Multiply $3$ by $- 1$ .

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

Step 2

Multiply $(x - 3) x^{- 3}$ .

$\int x \cdot x^{- 3} - 3 x^{- 3} d x$

Step 3

Multiply $x$ by $x^{- 3}$ by adding the exponents.

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Step 3.1

Multiply $x$ by $x^{- 3}$ .

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Step 3.1.1

Raise $x$ to the power of $1$ .

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

Step 3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2

Subtract $3$ from $1$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.2

Simplify.

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

Step 8.3

Simplify.

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Step 8.3.1

Multiply $- 1$ by $- 3$ .

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

Step 8.3.2

Combine $3$ and $\frac{1}{2 x^{2}}$ .

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