Calculus Examples

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

Step 1

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

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

Step 2

Simplify the expression.

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Step 2.1

Simplify.

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Step 2.1.1

Combine $d$ and $\frac{1}{4}$ .

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

Step 2.1.2

Combine $x$ and $\frac{d}{4}$ .

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

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

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

Step 4.3

Simplify.

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Step 4.3.1

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

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

Step 4.3.2

Combine $\frac{x}{4}$ and $d$ .

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

Step 4.3.3

Multiply $\frac{x d}{4}$ by $\frac{1}{2 x^{2}}$ .

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

Step 4.3.4

Multiply $2$ by $4$ .

$- \frac{x d}{8 x^{2}} + C$

Step 4.3.5

Cancel the common factor of $x$ and $x^{2}$ .

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Step 4.3.5.1

Factor $x$ out of $x d$ .

$- \frac{x (d)}{8 x^{2}} + C$

Step 4.3.5.2

Cancel the common factors.

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Step 4.3.5.2.1

Factor $x$ out of $8 x^{2}$ .

$- \frac{x (d)}{x (8 x)} + C$

Step 4.3.5.2.2

Cancel the common factor.

$- \frac{x d}{x (8 x)} + C$

Step 4.3.5.2.3

Rewrite the expression.

$- \frac{d}{8 x} + C$