Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Factor $x$ out of $5 x^{3} + 3 x^{2} + x$ .

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Step 1.1.1

Factor $x$ out of $5 x^{3}$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Factor $x$ out of $x (5 x^{2}) + x (3 x)$ .

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

Step 1.1.6

Factor $x$ out of $x (5 x^{2} + 3 x) + x \cdot 1$ .

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

Step 1.2

Cancel the common factors.

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 2

Divide $5 x^{2} + 3 x + 1$ by $x$ .

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Step 2.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$0$

$5 x^{2}$

$3 x$

$1$

Step 2.2

Divide the highest order term in the dividend $5 x^{2}$ by the highest order term in divisor $x$ .

					$5 x$
	$x$	+	$0$		$5 x^{2}$	+	$3 x$	+	$1$

Step 2.3

Multiply the new quotient term by the divisor.

				$5 x$
$x$	+	$0$		$5 x^{2}$	+	$3 x$	+	$1$
			+	$5 x^{2}$	+	$0$

Step 2.4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} + 0$

				$5 x$
$x$	+	$0$		$5 x^{2}$	+	$3 x$	+	$1$
			-	$5 x^{2}$	-	$0$

Step 2.5

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

				$5 x$
$x$	+	$0$		$5 x^{2}$	+	$3 x$	+	$1$
			-	$5 x^{2}$	-	$0$
					+	$3 x$

Step 2.6

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

				$5 x$
$x$	+	$0$		$5 x^{2}$	+	$3 x$	+	$1$
			-	$5 x^{2}$	-	$0$
					+	$3 x$	+	$1$

Step 2.7

Divide the highest order term in the dividend $3 x$ by the highest order term in divisor $x$ .

				$5 x$	+	$3$
$x$	+	$0$		$5 x^{2}$	+	$3 x$	+	$1$
			-	$5 x^{2}$	-	$0$
					+	$3 x$	+	$1$

Step 2.8

Multiply the new quotient term by the divisor.

				$5 x$	+	$3$
$x$	+	$0$		$5 x^{2}$	+	$3 x$	+	$1$
			-	$5 x^{2}$	-	$0$
					+	$3 x$	+	$1$
					+	$3 x$	+	$0$

Step 2.9

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

				$5 x$	+	$3$
$x$	+	$0$		$5 x^{2}$	+	$3 x$	+	$1$
			-	$5 x^{2}$	-	$0$
					+	$3 x$	+	$1$
					-	$3 x$	-	$0$

Step 2.10

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

				$5 x$	+	$3$
$x$	+	$0$		$5 x^{2}$	+	$3 x$	+	$1$
			-	$5 x^{2}$	-	$0$
					+	$3 x$	+	$1$
					-	$3 x$	-	$0$
							+	$1$

Step 2.11

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Reorder terms.

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