Calculus Examples

$\int_{1}^{e} \frac{4 x^{3} - 3 x}{x^{2}} d x$

Step 1

Simplify.

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Step 1.1

Factor $x$ out of $4 x^{3} - 3 x$ .

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Step 1.1.1

Factor $x$ out of $4 x^{3}$ .

$\int_{1}^{e} \frac{x (4 x^{2}) - 3 x}{x^{2}} d x$

Step 1.1.2

Factor $x$ out of $- 3 x$ .

$\int_{1}^{e} \frac{x (4 x^{2}) + x \cdot - 3}{x^{2}} d x$

Step 1.1.3

Factor $x$ out of $x (4 x^{2}) + x \cdot - 3$ .

$\int_{1}^{e} \frac{x (4 x^{2} - 3)}{x^{2}} d x$

Step 1.2

Cancel the common factors.

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Step 1.2.1

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

$\int_{1}^{e} \frac{x (4 x^{2} - 3)}{x \cdot x} d x$

Step 1.2.2

Cancel the common factor.

$\int_{1}^{e} \frac{x (4 x^{2} - 3)}{x \cdot x} d x$

Step 1.2.3

Rewrite the expression.

$\int_{1}^{e} \frac{4 x^{2} - 3}{x} d x$

Step 2

Divide $4 x^{2} - 3$ 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 $0$ .

$x$

$0$

$4 x^{2}$

$0 x$

$3$

Step 2.2

Divide the highest order term in the dividend $4 x^{2}$ by the highest order term in divisor $x$ .

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

Step 2.3

Multiply the new quotient term by the divisor.

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

Step 2.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{2} + 0$

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

$\int_{1}^{e} 4 x - \frac{3}{x} d x$

Step 3

Split the single integral into multiple integrals.

$\int_{1}^{e} 4 x d x + \int_{1}^{e} - \frac{3}{x} d x$

Step 4

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{1}^{e} x d x + \int_{1}^{e} - \frac{3}{x} d x$

Step 5

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

$4 (\frac{1}{2} x^{2}]_{1}^{e}) + \int_{1}^{e} - \frac{3}{x} d x$

Step 6

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

$4 (\frac{x^{2}}{2}]_{1}^{e}) + \int_{1}^{e} - \frac{3}{x} d x$

Step 7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$4 (\frac{x^{2}}{2}]_{1}^{e}) - \int_{1}^{e} \frac{3}{x} d x$

Step 8

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

$4 (\frac{x^{2}}{2}]_{1}^{e}) - (3 \int_{1}^{e} \frac{1}{x} d x)$

Step 9

Multiply $3$ by $- 1$ .

$4 (\frac{x^{2}}{2}]_{1}^{e}) - 3 \int_{1}^{e} \frac{1}{x} d x$

Step 10

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

$4 (\frac{x^{2}}{2}]_{1}^{e}) - 3 (\ln (| x |)]_{1}^{e})$

Step 11

Simplify the answer.

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Step 11.1

Substitute and simplify.

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Step 11.1.1

Evaluate $\frac{x^{2}}{2}$ at $e$ and at $1$ .

$4 ((\frac{e^{2}}{2}) - \frac{1^{2}}{2}) - 3 (\ln (| x |)]_{1}^{e})$

Step 11.1.2

Evaluate $\ln (| x |)$ at $e$ and at $1$ .

$4 (\frac{e^{2}}{2} - \frac{1^{2}}{2}) - 3 (\ln (| e |) - \ln (| 1 |))$

Step 11.1.3

One to any power is one.

$4 (\frac{e^{2}}{2} - \frac{1}{2}) - 3 (\ln (| e |) - \ln (| 1 |))$

Step 11.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$4 (\frac{e^{2}}{2} - \frac{1}{2}) - 3 \ln (\frac{| e |}{| 1 |})$

Step 11.3

Simplify.

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Step 11.3.1

Apply the distributive property.

$4 \frac{e^{2}}{2} + 4 (- \frac{1}{2}) - 3 \ln (\frac{| e |}{| 1 |})$

Step 11.3.2

Cancel the common factor of $2$ .

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Step 11.3.2.1

Factor $2$ out of $4$ .

$2 (2) \frac{e^{2}}{2} + 4 (- \frac{1}{2}) - 3 \ln (\frac{| e |}{| 1 |})$

Step 11.3.2.2

Cancel the common factor.

$2 \cdot 2 \frac{e^{2}}{2} + 4 (- \frac{1}{2}) - 3 \ln (\frac{| e |}{| 1 |})$

Step 11.3.2.3

Rewrite the expression.

$2 e^{2} + 4 (- \frac{1}{2}) - 3 \ln (\frac{| e |}{| 1 |})$

Step 11.3.3

Cancel the common factor of $2$ .

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Step 11.3.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$2 e^{2} + 4 (\frac{- 1}{2}) - 3 \ln (\frac{| e |}{| 1 |})$

Step 11.3.3.2

Factor $2$ out of $4$ .

$2 e^{2} + 2 (2) \frac{- 1}{2} - 3 \ln (\frac{| e |}{| 1 |})$

Step 11.3.3.3

Cancel the common factor.

$2 e^{2} + 2 \cdot 2 \frac{- 1}{2} - 3 \ln (\frac{| e |}{| 1 |})$

Step 11.3.3.4

Rewrite the expression.

$2 e^{2} + 2 \cdot - 1 - 3 \ln (\frac{| e |}{| 1 |})$

Step 11.3.4

Multiply $2$ by $- 1$ .

$2 e^{2} - 2 - 3 \ln (\frac{| e |}{| 1 |})$

Step 11.3.5

$e$ is approximately $2.71828182$ which is positive so remove the absolute value

$2 e^{2} - 2 - 3 \ln (\frac{e}{| 1 |})$

Step 11.3.6

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$2 e^{2} - 2 - 3 \ln (\frac{e}{1})$

Step 11.3.7

Divide $e$ by $1$ .

$2 e^{2} - 2 - 3 \ln (e)$

Step 11.3.8

The natural logarithm of $e$ is $1$ .

$2 e^{2} - 2 - 3 \cdot 1$

Step 11.3.9

Multiply $- 3$ by $1$ .

$2 e^{2} - 2 - 3$

Step 11.3.10

Subtract $3$ from $- 2$ .

$2 e^{2} - 5$

Step 12

The result can be shown in multiple forms.

Exact Form:

$2 e^{2} - 5$

Decimal Form:

$9.77811219 \dots$

Step 13