Calculus Examples

$\int \frac{x^{2} - 3 x - 5}{x} d x$

Step 1

Split the fraction into multiple fractions.

$\int \frac{x^{2}}{x} + \frac{- 3 x}{x} + \frac{- 5}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{x^{2}}{x} d x + \int \frac{- 3 x}{x} d x + \int \frac{- 5}{x} d x$

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

$\int \frac{x \cdot x}{x} d x + \int \frac{- 3 x}{x} d x + \int \frac{- 5}{x} d x$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

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

$\int \frac{x \cdot x}{x^{1}} d x + \int \frac{- 3 x}{x} d x + \int \frac{- 5}{x} d x$

Step 3.1.2.2

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

$\int \frac{x \cdot x}{x \cdot 1} d x + \int \frac{- 3 x}{x} d x + \int \frac{- 5}{x} d x$

Step 3.1.2.3

Cancel the common factor.

$\int \frac{x \cdot x}{x \cdot 1} d x + \int \frac{- 3 x}{x} d x + \int \frac{- 5}{x} d x$

Step 3.1.2.4

Rewrite the expression.

$\int \frac{x}{1} d x + \int \frac{- 3 x}{x} d x + \int \frac{- 5}{x} d x$

Step 3.1.2.5

Divide $x$ by $1$ .

$\int x d x + \int \frac{- 3 x}{x} d x + \int \frac{- 5}{x} d x$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Cancel the common factor.

$\int x d x + \int \frac{- 3 x}{x} d x + \int \frac{- 5}{x} d x$

Step 3.2.2

Divide $- 3$ by $1$ .

$\int x d x + \int - 3 d x + \int \frac{- 5}{x} d x$

Step 4

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

$\frac{1}{2} x^{2} + C + \int - 3 d x + \int \frac{- 5}{x} d x$

Step 5

Move the negative in front of the fraction.

$\frac{1}{2} x^{2} + C + \int - 3 d x + \int - \frac{5}{x} d x$

Step 6

Apply the constant rule.

$\frac{1}{2} x^{2} + C - 3 x + C + \int - \frac{5}{x} d x$

Step 7

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

$\frac{1}{2} x^{2} + C - 3 x + C - \int \frac{5}{x} d x$

Step 8

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\frac{1}{2} x^{2} + C - 3 x + C - (5 \int \frac{1}{x} d x)$

Step 9

Multiply $5$ by $- 1$ .

$\frac{1}{2} x^{2} + C - 3 x + C - 5 \int \frac{1}{x} d x$

Step 10

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

$\frac{1}{2} x^{2} + C - 3 x + C - 5 (\ln (| x |) + C)$

Step 11

Simplify.

$\frac{1}{2} x^{2} - 3 x - 5 \ln (| x |) + C$