Calculus Examples

$\frac{x}{x^{2}}$

Step 1

Write $\frac{x}{x^{2}}$ as a function.

$f (x) = \frac{x}{x^{2}}$

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.

$F (x) = \int \frac{x}{x^{2}} d x$

Step 4

Cancel the common factor of $x$ and $x^{2}$ .

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Step 4.1

Raise $x$ to the power of $1$ .

$\int \frac{x^{1}}{x^{2}} d x$

Step 4.2

Factor $x$ out of $x^{1}$ .

$\int \frac{x \cdot 1}{x^{2}} d x$

Step 4.3

Cancel the common factors.

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Step 4.3.1

Factor $x$ out of $x^{2}$ .

$\int \frac{x \cdot 1}{x \cdot x} d x$

Step 4.3.2

Cancel the common factor.

$\int \frac{x \cdot 1}{x \cdot x} d x$

Step 4.3.3

Rewrite the expression.

$\int \frac{1}{x} d x$

Step 5

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

$\ln (| x |) + C$

Step 6

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

$F (x) =$ $\ln (| x |) + C$