Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 3

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

Tap for more steps...

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 $0$ .

$x$

$0$

$3 x^{2}$

$4 x$

$1$

Step 3.2

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

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

Step 3.3

Multiply the new quotient term by the divisor.

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

Step 3.4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} + 0$

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $x$ .

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

Step 3.8

Multiply the new quotient term by the divisor.

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

Step 3.9

The expression needs to be subtracted from the dividend, so change all the signs in $4 x + 0$

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

Step 3.10

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

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

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.

Tap for more steps...

Step 10.1

Simplify.

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

Step 10.2

Reorder terms.

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