Calculus Examples

$f (x) = 5 x^{- 3} + 2 x - y$

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.

$F (x) = \int 5 x^{- 3} + 2 x - y d x$

Step 3

Split the single integral into multiple integrals.

$\int 5 x^{- 3} d x + \int 2 x d x + \int - y d x$

Step 4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int x^{- 3} d x + \int 2 x d x + \int - y d x$

Step 5

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

$5 (- \frac{1}{2} x^{- 2} + C) + \int 2 x d x + \int - y d x$

Step 6

Simplify.

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Step 6.1

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

$5 (- \frac{x^{- 2}}{2} + C) + \int 2 x d x + \int - y d x$

Step 6.2

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

$5 (- \frac{1}{2 x^{2}} + C) + \int 2 x d x + \int - y d x$

Step 7

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

$5 (- \frac{1}{2 x^{2}} + C) + 2 \int x d x + \int - y d x$

Step 8

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

$5 (- \frac{1}{2 x^{2}} + C) + 2 (\frac{1}{2} x^{2} + C) + \int - y d x$

Step 9

Apply the constant rule.

$5 (- \frac{1}{2 x^{2}} + C) + 2 (\frac{1}{2} x^{2} + C) - y x + C$

Step 10

Simplify.

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Step 10.1

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

$5 (- \frac{1}{2 x^{2}} + C) + 2 (\frac{x^{2}}{2} + C) - y x + C$

Step 10.2

Simplify.

$- \frac{5}{2 x^{2}} + x^{2} - y x + C$

Step 11

The answer is the antiderivative of the function $f (x) = 5 x^{- 3} + 2 x - y$ .

$F (x) =$ $- \frac{5}{2 x^{2}} + x^{2} - y x + C$