Calculus Examples

$\int \frac{x^{3} - 7}{x} d x$

Step 1

Split the fraction into multiple fractions.

$\int \frac{x^{3}}{x} + \frac{- 7}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{x^{3}}{x} d x + \int \frac{- 7}{x} d x$

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x^{3}$ and $x$ .

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Step 3.1.1

Factor $x$ out of $x^{3}$ .

$\int \frac{x \cdot x^{2}}{x} d x + \int \frac{- 7}{x} d x$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

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

$\int \frac{x \cdot x^{2}}{x^{1}} d x + \int \frac{- 7}{x} d x$

Step 3.1.2.2

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

$\int \frac{x \cdot x^{2}}{x \cdot 1} d x + \int \frac{- 7}{x} d x$

Step 3.1.2.3

Cancel the common factor.

$\int \frac{x \cdot x^{2}}{x \cdot 1} d x + \int \frac{- 7}{x} d x$

Step 3.1.2.4

Rewrite the expression.

$\int \frac{x^{2}}{1} d x + \int \frac{- 7}{x} d x$

Step 3.1.2.5

Divide $x^{2}$ by $1$ .

$\int x^{2} d x + \int \frac{- 7}{x} d x$

Step 3.2

Move the negative in front of the fraction.

$\int x^{2} d x + \int - \frac{7}{x} d x$

Step 4

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

$\frac{1}{3} x^{3} + C + \int - \frac{7}{x} d x$

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{3} x^{3} + C - \int \frac{7}{x} d x$

Step 6

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$\frac{1}{3} x^{3} + C - (7 \int \frac{1}{x} d x)$

Step 7

Multiply $7$ by $- 1$ .

$\frac{1}{3} x^{3} + C - 7 \int \frac{1}{x} d x$

Step 8

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

$\frac{1}{3} x^{3} + C - 7 (\ln (| x |) + C)$

Step 9

Simplify.

$\frac{1}{3} x^{3} - 7 \ln (| x |) + C$