Calculus Examples

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

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $3$ by $- 1$ .

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

Step 3

Expand $(x^{3} - 4 x - 1) x^{- 3}$ .

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

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

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

Step 3.4

Subtract $3$ from $3$ .

$\frac{1}{2} \int x^{0} - 4 x \cdot x^{- 3} - 1 x^{- 3} d x$

Step 3.5

Anything raised to $0$ is $1$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Subtract $3$ from $1$ .

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

Step 3.9

Reorder $1$ and $- 4 x^{- 2}$ .

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

Step 3.10

Move $1$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

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

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

Step 10

Apply the constant rule.

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

Step 11

Simplify.

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