Calculus Examples

$\int \frac{6 + x^{2}}{x} d x$

Step 1

Split the fraction into multiple fractions.

$\int \frac{6}{x} + \frac{x^{2}}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{6}{x} d x + \int \frac{x^{2}}{x} d x$

Step 3

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

Tap for more steps...

Step 3.1

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

$\int \frac{6}{x} d x + \int \frac{x \cdot x}{x} d x$

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

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

$\int \frac{6}{x} d x + \int \frac{x \cdot x}{x^{1}} d x$

Step 3.2.2

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

$\int \frac{6}{x} d x + \int \frac{x \cdot x}{x \cdot 1} d x$

Step 3.2.3

Cancel the common factor.

$\int \frac{6}{x} d x + \int \frac{x \cdot x}{x \cdot 1} d x$

Step 3.2.4

Rewrite the expression.

$\int \frac{6}{x} d x + \int \frac{x}{1} d x$

Step 3.2.5

Divide $x$ by $1$ .

$\int \frac{6}{x} d x + \int x d x$

Step 4

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \frac{1}{x} d x + \int x d x$

Step 5

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

$6 (\ln (| x |) + C) + \int x d x$

Step 6

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

$6 (\ln (| x |) + C) + \frac{1}{2} x^{2} + C$

Step 7

Simplify.

$6 \ln (| x |) + \frac{1}{2} x^{2} + C$