Calculus Examples

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

Step 1

Remove parentheses.

$\int \frac{3 x^{3} - 4 x - 2}{x^{3}} d x$

Step 2

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

$\int (3 x^{3} - 4 x - 2) {(x^{3})}^{- 1} d x$

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $3$ by $- 1$ .

$\int (3 x^{3} - 4 x - 2) x^{- 3} d x$

Step 4

Expand $(3 x^{3} - 4 x - 2) x^{- 3}$ .

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Step 4.1

Apply the distributive property.

$\int (3 x^{3} - 4 x) x^{- 3} - 2 x^{- 3} d x$

Step 4.2

Apply the distributive property.

$\int 3 x^{3} x^{- 3} - 4 x \cdot x^{- 3} - 2 x^{- 3} d x$

Step 4.3

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

$\int 3 x^{3 - 3} - 4 x \cdot x^{- 3} - 2 x^{- 3} d x$

Step 4.4

Subtract $3$ from $3$ .

$\int 3 x^{0} - 4 x \cdot x^{- 3} - 2 x^{- 3} d x$

Step 4.5

Anything raised to $0$ is $1$ .

$\int 3 \cdot 1 - 4 x \cdot x^{- 3} - 2 x^{- 3} d x$

Step 4.6

Multiply $3$ by $1$ .

$\int 3 - 4 x \cdot x^{- 3} - 2 x^{- 3} d x$

Step 4.7

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

$\int 3 - 4 (x^{1} x^{- 3}) - 2 x^{- 3} d x$

Step 4.8

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

$\int 3 - 4 x^{1 - 3} - 2 x^{- 3} d x$

Step 4.9

Subtract $3$ from $1$ .

$\int 3 - 4 x^{- 2} - 2 x^{- 3} d x$

Step 4.10

Reorder $3$ and $- 4 x^{- 2}$ .

$\int - 4 x^{- 2} + 3 - 2 x^{- 3} d x$

Step 4.11

Move $3$ .

$\int - 4 x^{- 2} - 2 x^{- 3} + 3 d x$

Step 5

Split the single integral into multiple integrals.

$\int - 4 x^{- 2} d x + \int - 2 x^{- 3} d x + \int 3 d x$

Step 6

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$- 4 \int x^{- 2} d x + \int - 2 x^{- 3} d x + \int 3 d x$

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

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

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

Step 11

Apply the constant rule.

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

Step 12

Simplify.

$\frac{4}{x} + \frac{1}{x^{2}} + 3 x + C$