Calculus Examples

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

Step 1

Write $\frac{1}{x^{2}} d x$ as a function.

$f (x) = \frac{1}{x^{2}} d x$

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.

$F (x) = \int \frac{1}{x^{2}} \cdot (d x) d x$

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

Factor $x$ out of $d x$ .

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

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

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 5

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

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

Step 6

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

$d (\ln (| x |) + C)$

Step 7

Simplify.

$d \ln (| x |) + C$

Step 8

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

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