Calculus Examples

$x^{2} + \frac{3}{x} - 2 \sqrt{x}$

Step 1

Write $x^{2} + \frac{3}{x} - 2 \sqrt{x}$ as a function.

$f (x) = x^{2} + \frac{3}{x} - 2 \sqrt{x}$

Step 2

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 3

Set up the integral to solve.

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

Step 4

Split the single integral into multiple integrals.

$\int x^{2} d x + \int \frac{3}{x} d x + \int - 2 \sqrt{x} d x$

Step 5

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

$\frac{1}{3} x^{3} + C + \int \frac{3}{x} d x + \int - 2 \sqrt{x} d x$

Step 6

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

$\frac{1}{3} x^{3} + C + 3 \int \frac{1}{x} d x + \int - 2 \sqrt{x} d x$

Step 7

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{1}{3} x^{3} + C + 3 (\ln (| x |) + C) + \int - 2 \sqrt{x} d x$

Step 8

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

$\frac{1}{3} x^{3} + C + 3 (\ln (| x |) + C) - 2 \int \sqrt{x} d x$

Step 9

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

$\frac{1}{3} x^{3} + C + 3 (\ln (| x |) + C) - 2 \int x^{\frac{1}{2}} d x$

Step 10

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

$\frac{1}{3} x^{3} + C + 3 (\ln (| x |) + C) - 2 (\frac{2}{3} x^{\frac{3}{2}} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

$\frac{1}{3} x^{3} + 3 \ln (| x |) - 2 (\frac{2}{3}) x^{\frac{3}{2}} + C$

Step 11.2

Simplify.

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Step 11.2.1

Combine $- 2$ and $\frac{2}{3}$ .

$\frac{1}{3} x^{3} + 3 \ln (| x |) + \frac{- 2 \cdot 2}{3} x^{\frac{3}{2}} + C$

Step 11.2.2

Multiply $- 2$ by $2$ .

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

Step 11.2.3

Move the negative in front of the fraction.

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

Step 12

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

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