Calculus Examples

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

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Remove parentheses.

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

Step 3

Split the fraction into multiple fractions.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Simplify.

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Step 5.1

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

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Step 5.1.1

Factor $x$ out of $3 x^{2}$ .

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

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

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

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

Step 5.1.2.2

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

$\int \frac{x (3 x)}{x \cdot 1} d x + \int \frac{- 2 x}{x} d x + \int \frac{1}{x} d x$

Step 5.1.2.3

Cancel the common factor.

$\int \frac{x (3 x)}{x \cdot 1} d x + \int \frac{- 2 x}{x} d x + \int \frac{1}{x} d x$

Step 5.1.2.4

Rewrite the expression.

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

Step 5.1.2.5

Divide $3 x$ by $1$ .

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

Step 5.2

Cancel the common factor of $x$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Divide $- 2$ by $1$ .

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

Step 6

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 7

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

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

Step 8

Apply the constant rule.

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Reorder terms.

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