Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Split the fraction into multiple fractions.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Simplify.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

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

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

Step 6.1.2.2

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

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

Step 6.1.2.3

Cancel the common factor.

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

Step 6.1.2.4

Rewrite the expression.

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

Step 6.1.2.5

Divide $x$ by $1$ .

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

Step 6.2

Cancel the common factor of $x$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 7

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

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

Step 8

Apply the constant rule.

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

Step 9

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

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

Step 10

Simplify.

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

Step 11

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

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