Calculus Examples

\int (\frac{3 x^{2} + 4 x + 1}{2 x}) d x

$\int (\frac{3 x^{2} + 4 x + 1}{2 x}) d x$

Step 1

Remove parentheses.

\int \frac{3 x^{2} + 4 x + 1}{2 x} d x

$\int \frac{3 x^{2} + 4 x + 1}{2 x} d x$

Step 2

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

x

$x$ , move

\frac{1}{2}

$\frac{1}{2}$ out of the integral.

\frac{1}{2} \int \frac{3 x^{2} + 4 x + 1}{x} d x

$\frac{1}{2} \int \frac{3 x^{2} + 4 x + 1}{x} d x$

Step 3

Divide

3 x^{2} + 4 x + 1

$3 x^{2} + 4 x + 1$ by

x

$x$ .

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Step 3.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of

0

$0$ .

x

$x$

0

$0$

3 x^{2}

$3 x^{2}$

4 x

$4 x$

1

$1$

Step 3.2

Divide the highest order term in the dividend

3 x^{2}

$3 x^{2}$ by the highest order term in divisor

x

$x$ .

					$3 x$ $3 x$
	$x$ $x$	+	$0$ $0$		$3 x^{2}$ $3 x^{2}$	+	$4 x$ $4 x$	+	$1$ $1$

Step 3.3

Multiply the new quotient term by the divisor.

				$3 x$ $3 x$
$x$ $x$	+	$0$ $0$		$3 x^{2}$ $3 x^{2}$	+	$4 x$ $4 x$	+	$1$ $1$
			+	$3 x^{2}$ $3 x^{2}$	+	$0$ $0$

Step 3.4

The expression needs to be subtracted from the dividend, so change all the signs in

3 x^{2} + 0

$3 x^{2} + 0$

				$3 x$ $3 x$
$x$ $x$	+	$0$ $0$		$3 x^{2}$ $3 x^{2}$	+	$4 x$ $4 x$	+	$1$ $1$
			-	$3 x^{2}$ $3 x^{2}$	-	$0$ $0$

Step 3.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$3 x$ $3 x$
$x$ $x$	+	$0$ $0$		$3 x^{2}$ $3 x^{2}$	+	$4 x$ $4 x$	+	$1$ $1$
			-	$3 x^{2}$ $3 x^{2}$	-	$0$ $0$
					+	$4 x$ $4 x$

Step 3.6

Pull the next terms from the original dividend down into the current dividend.

				$3 x$ $3 x$
$x$ $x$	+	$0$ $0$		$3 x^{2}$ $3 x^{2}$	+	$4 x$ $4 x$	+	$1$ $1$
			-	$3 x^{2}$ $3 x^{2}$	-	$0$ $0$
					+	$4 x$ $4 x$	+	$1$ $1$

Step 3.7

Divide the highest order term in the dividend

4 x

$4 x$ by the highest order term in divisor

x

$x$ .

				$3 x$ $3 x$	+	$4$ $4$
$x$ $x$	+	$0$ $0$		$3 x^{2}$ $3 x^{2}$	+	$4 x$ $4 x$	+	$1$ $1$
			-	$3 x^{2}$ $3 x^{2}$	-	$0$ $0$
					+	$4 x$ $4 x$	+	$1$ $1$

Step 3.8

Multiply the new quotient term by the divisor.

				$3 x$ $3 x$	+	$4$ $4$
$x$ $x$	+	$0$ $0$		$3 x^{2}$ $3 x^{2}$	+	$4 x$ $4 x$	+	$1$ $1$
			-	$3 x^{2}$ $3 x^{2}$	-	$0$ $0$
					+	$4 x$ $4 x$	+	$1$ $1$
					+	$4 x$ $4 x$	+	$0$ $0$

Step 3.9

The expression needs to be subtracted from the dividend, so change all the signs in

4 x + 0

$4 x + 0$

				$3 x$ $3 x$	+	$4$ $4$
$x$ $x$	+	$0$ $0$		$3 x^{2}$ $3 x^{2}$	+	$4 x$ $4 x$	+	$1$ $1$
			-	$3 x^{2}$ $3 x^{2}$	-	$0$ $0$
					+	$4 x$ $4 x$	+	$1$ $1$
					-	$4 x$ $4 x$	-	$0$ $0$

Step 3.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$3 x$ $3 x$	+	$4$ $4$
$x$ $x$	+	$0$ $0$		$3 x^{2}$ $3 x^{2}$	+	$4 x$ $4 x$	+	$1$ $1$
			-	$3 x^{2}$ $3 x^{2}$	-	$0$ $0$
					+	$4 x$ $4 x$	+	$1$ $1$
					-	$4 x$	-	$0$
							+	$1$

Step 3.11

The final answer is the quotient plus the remainder over the divisor.

$\frac{1}{2} \int 3 x + 4 + \frac{1}{x} d x$

Step 4

Split the single integral into multiple integrals.

$\frac{1}{2} (\int 3 x d x + \int 4 d x + \int \frac{1}{x} d x)$

Step 5

Since

$3$ is constant with respect to

$x$ , move

$3$ out of the integral.

$\frac{1}{2} (3 \int x d x + \int 4 d x + \int \frac{1}{x} d x)$

Step 6

By the Power Rule, the integral of

$x$ with respect to

$x$ is

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

$\frac{1}{2} (3 (\frac{1}{2} x^{2} + C) + \int 4 d x + \int \frac{1}{x} d x)$

Step 7

Apply the constant rule.

$\frac{1}{2} (3 (\frac{1}{2} x^{2} + C) + 4 x + C + \int \frac{1}{x} d x)$

Step 8

Combine

$\frac{1}{2}$ and

$x^{2}$ .

$\frac{1}{2} (3 (\frac{x^{2}}{2} + C) + 4 x + C + \int \frac{1}{x} d x)$

Step 9

The integral of

$\frac{1}{x}$ with respect to

$x$ is

$\ln (| x |)$ .

$\frac{1}{2} (3 (\frac{x^{2}}{2} + C) + 4 x + C + \ln (| x |) + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{1}{2} (\frac{3 x^{2}}{2} + 4 x + \ln (| x |)) + C$

Step 10.2

Reorder terms.

$\frac{1}{2} (\frac{3}{2} x^{2} + 4 x + \ln (| x |)) + C$