Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

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

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

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

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

Step 3.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 3.2.1

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

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

Step 3.2.2

Multiply $4$ by $- 1$ .

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

Step 4

By the Power Rule, the integral of $x^{- 4}$ with respect to $x$ is $- \frac{1}{3} x^{- 3}$ .

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

Step 5

Simplify the answer.

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Step 5.1

Simplify.

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Step 5.1.1

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

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

Step 5.1.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.2

Simplify.

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

Step 5.3

Simplify.

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Step 5.3.1

Multiply $- 1$ by $3$ .

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

Step 5.3.2

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

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

Step 5.3.3

Cancel the common factor of $- 3$ and $3$ .

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Step 5.3.3.1

Factor $3$ out of $- 3$ .

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

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $3$ out of $3 x^{3}$ .

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

Step 5.3.3.2.2

Cancel the common factor.

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

Step 5.3.3.2.3

Rewrite the expression.

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

Step 5.3.4

Move the negative in front of the fraction.

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