Calculus Examples

$x^{2} - 3 x + \frac{1}{x}$

Step 1

Write $x^{2} - 3 x + \frac{1}{x}$ as a function.

$f (x) = x^{2} - 3 x + \frac{1}{x}$

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.

$F (x) = \int x^{2} - 3 x + \frac{1}{x} d x$

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Simplify.

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

Step 10

Reorder terms.

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

Step 11

The answer is the antiderivative of the function $f (x) = x^{2} - 3 x + \frac{1}{x}$ .

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