Calculus Examples

$f (x) = \frac{3}{x^{4} y}$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

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

Step 3

Since $\frac{3}{y}$ is constant with respect to $x$ , move $\frac{3}{y}$ out of the integral.

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $4$ by $- 1$ .

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify.

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.2

Simplify.

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

Step 6.3

Simplify.

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Step 6.3.1

Multiply $\frac{3}{y}$ by $\frac{1}{3 x^{3}}$ .

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

Step 6.3.2

Move $3$ to the left of $y$ .

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

Step 6.3.3

Cancel the common factor of $3$ .

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Step 6.3.3.1

Cancel the common factor.

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

Step 6.3.3.2

Rewrite the expression.

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

Step 7

The answer is the antiderivative of the function $f (x) = \frac{3}{x^{4} y}$ .

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