Calculus Examples

$f (x) = \frac{1}{x^{- 3}} + 5 x y - \sqrt{x}$

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.

$F (x) = \int \frac{1}{x^{- 3}} + 5 x y - \sqrt{x} d x$

Step 3

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

$\frac{1}{4} x^{4} + C + \int 5 x y d x + \int - \sqrt{x} d x$

Step 6

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

$\frac{1}{4} x^{4} + C + 5 y \int x d x + \int - \sqrt{x} d x$

Step 7

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

$\frac{1}{4} x^{4} + C + 5 y (\frac{1}{2} x^{2} + C) + \int - \sqrt{x} d x$

Step 8

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

$\frac{1}{4} x^{4} + C + 5 y (\frac{1}{2} x^{2} + C) - \int \sqrt{x} d x$

Step 9

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{1}{4} x^{4} + C + 5 y (\frac{1}{2} x^{2} + C) - \int x^{\frac{1}{2}} d x$

Step 10

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

$\frac{1}{4} x^{4} + C + 5 y (\frac{1}{2} x^{2} + C) - (\frac{2}{3} x^{\frac{3}{2}} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

$\frac{x^{4}}{4} + \frac{5 y x^{2}}{2} - (\frac{2}{3} x^{\frac{3}{2}}) + C$

Step 11.2

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$\frac{x^{4}}{4} + \frac{5 y x^{2}}{2} - \frac{2 x^{\frac{3}{2}}}{3} + C$

Step 12

Reorder terms.

$\frac{1}{4} x^{4} + \frac{5}{2} y x^{2} - \frac{2}{3} x^{\frac{3}{2}} + C$

Step 13

The answer is the antiderivative of the function $f (x) = \frac{1}{x^{- 3}} + 5 x y - \sqrt{x}$ .

$F (x) =$ $\frac{1}{4} x^{4} + \frac{5}{2} y x^{2} - \frac{2}{3} x^{\frac{3}{2}} + C$