Calculus Examples

$\int \frac{2 x^{2} + 7 x - 3}{x - 2} d x$

Step 1

Let $u = x - 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x - 2$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x - 2$ .

$\frac{d}{d x} [x - 2]$

Step 1.1.2

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 2]$

Step 1.1.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{2 {(u + 2)}^{2} + 7 (u + 2) - 3}{u} d u$

Step 2

Split the fraction into multiple fractions.

$\int \frac{2 {(u + 2)}^{2}}{u} + \frac{7 (u + 2)}{u} + \frac{- 3}{u} d u$

Step 3

Split the single integral into multiple integrals.

$\int \frac{2 {(u + 2)}^{2}}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int \frac{- 3}{u} d u$

Step 4

Move the negative in front of the fraction.

$\int \frac{2 {(u + 2)}^{2}}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 5

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 \int \frac{{(u + 2)}^{2}}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 6

Simplify by multiplying through.

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Step 6.1

Rewrite ${(u + 2)}^{2}$ as $(u + 2) (u + 2)$ .

$2 \int \frac{(u + 2) (u + 2)}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 6.2

Apply the distributive property.

$2 \int \frac{u (u + 2) + 2 (u + 2)}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 6.3

Apply the distributive property.

$2 \int \frac{u \cdot u + u \cdot 2 + 2 (u + 2)}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 6.4

Apply the distributive property.

$2 \int \frac{u \cdot u + u \cdot 2 + 2 u + 2 \cdot 2}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 6.5

Reorder $u$ and $2$ .

$2 \int \frac{u \cdot u + 2 \cdot u + 2 u + 2 \cdot 2}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 7

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

$2 \int \frac{u^{1} u + 2 u + 2 u + 2 \cdot 2}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 8

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

$2 \int \frac{u^{1} u^{1} + 2 u + 2 u + 2 \cdot 2}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 \int \frac{u^{1 + 1} + 2 u + 2 u + 2 \cdot 2}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 10

Simplify the expression.

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Step 10.1

Add $1$ and $1$ .

$2 \int \frac{u^{2} + 2 u + 2 u + 2 \cdot 2}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 10.2

Multiply $2$ by $2$ .

$2 \int \frac{u^{2} + 2 u + 2 u + 4}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 11

Add $2 u$ and $2 u$ .

$2 \int \frac{u^{2} + 4 u + 4}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 12

Divide $u^{2} + 4 u + 4$ by $u$ .

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Step 12.1

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

$u$

$0$

$u^{2}$

$4 u$

$4$

Step 12.2

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

					$u$
	$u$	+	$0$		$u^{2}$	+	$4 u$	+	$4$

Step 12.3

Multiply the new quotient term by the divisor.

				$u$
$u$	+	$0$		$u^{2}$	+	$4 u$	+	$4$
			+	$u^{2}$	+	$0$

Step 12.4

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

				$u$
$u$	+	$0$		$u^{2}$	+	$4 u$	+	$4$
			-	$u^{2}$	-	$0$

Step 12.5

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

				$u$
$u$	+	$0$		$u^{2}$	+	$4 u$	+	$4$
			-	$u^{2}$	-	$0$
					+	$4 u$

Step 12.6

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

				$u$
$u$	+	$0$		$u^{2}$	+	$4 u$	+	$4$
			-	$u^{2}$	-	$0$
					+	$4 u$	+	$4$

Step 12.7

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

				$u$	+	$4$
$u$	+	$0$		$u^{2}$	+	$4 u$	+	$4$
			-	$u^{2}$	-	$0$
					+	$4 u$	+	$4$

Step 12.8

Multiply the new quotient term by the divisor.

				$u$	+	$4$
$u$	+	$0$		$u^{2}$	+	$4 u$	+	$4$
			-	$u^{2}$	-	$0$
					+	$4 u$	+	$4$
					+	$4 u$	+	$0$

Step 12.9

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

				$u$	+	$4$
$u$	+	$0$		$u^{2}$	+	$4 u$	+	$4$
			-	$u^{2}$	-	$0$
					+	$4 u$	+	$4$
					-	$4 u$	-	$0$

Step 12.10

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

				$u$	+	$4$
$u$	+	$0$		$u^{2}$	+	$4 u$	+	$4$
			-	$u^{2}$	-	$0$
					+	$4 u$	+	$4$
					-	$4 u$	-	$0$
							+	$4$

Step 12.11

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

$2 \int u + 4 + \frac{4}{u} d u + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 13

Split the single integral into multiple integrals.

$2 (\int u d u + \int 4 d u + \int \frac{4}{u} d u) + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 14

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

$2 (\frac{1}{2} u^{2} + C + \int 4 d u + \int \frac{4}{u} d u) + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 15

Apply the constant rule.

$2 (\frac{1}{2} u^{2} + C + 4 u + C + \int \frac{4}{u} d u) + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 16

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

$2 (\frac{u^{2}}{2} + C + 4 u + C + \int \frac{4}{u} d u) + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 17

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

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 \int \frac{1}{u} d u) + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 18

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

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 (\ln (| u |) + C)) + \int \frac{7 (u + 2)}{u} d u + \int - \frac{3}{u} d u$

Step 19

Since $7$ is constant with respect to $u$ , move $7$ out of the integral.

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 (\ln (| u |) + C)) + 7 \int \frac{u + 2}{u} d u + \int - \frac{3}{u} d u$

Step 20

Divide $u + 2$ by $u$ .

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Step 20.1

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

$u$

$0$

$u$

$2$

Step 20.2

Divide the highest order term in the dividend $u$ by the highest order term in divisor $u$ .

					$1$
	$u$	+	$0$		$u$	+	$2$

Step 20.3

Multiply the new quotient term by the divisor.

				$1$
$u$	+	$0$		$u$	+	$2$
			+	$u$	+	$0$

Step 20.4

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

				$1$
$u$	+	$0$		$u$	+	$2$
			-	$u$	-	$0$

Step 20.5

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

				$1$
$u$	+	$0$		$u$	+	$2$
			-	$u$	-	$0$
					+	$2$

Step 20.6

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

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 (\ln (| u |) + C)) + 7 \int 1 + \frac{2}{u} d u + \int - \frac{3}{u} d u$

Step 21

Split the single integral into multiple integrals.

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 (\ln (| u |) + C)) + 7 (\int d u + \int \frac{2}{u} d u) + \int - \frac{3}{u} d u$

Step 22

Apply the constant rule.

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 (\ln (| u |) + C)) + 7 (u + C + \int \frac{2}{u} d u) + \int - \frac{3}{u} d u$

Step 23

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 (\ln (| u |) + C)) + 7 (u + C + 2 \int \frac{1}{u} d u) + \int - \frac{3}{u} d u$

Step 24

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

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 (\ln (| u |) + C)) + 7 (u + C + 2 (\ln (| u |) + C)) + \int - \frac{3}{u} d u$

Step 25

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

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 (\ln (| u |) + C)) + 7 (u + C + 2 (\ln (| u |) + C)) - \int \frac{3}{u} d u$

Step 26

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

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 (\ln (| u |) + C)) + 7 (u + C + 2 (\ln (| u |) + C)) - (3 \int \frac{1}{u} d u)$

Step 27

Multiply $3$ by $- 1$ .

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 (\ln (| u |) + C)) + 7 (u + C + 2 (\ln (| u |) + C)) - 3 \int \frac{1}{u} d u$

Step 28

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

$2 (\frac{u^{2}}{2} + C + 4 u + C + 4 (\ln (| u |) + C)) + 7 (u + C + 2 (\ln (| u |) + C)) - 3 (\ln (| u |) + C)$

Step 29

Simplify.

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Step 29.1

Simplify.

$u^{2} + 8 u + 8 \ln (| u |) + 7 u + 14 \ln (| u |) - 3 \ln (| u |) + C$

Step 29.2

Simplify.

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Step 29.2.1

Add $8 u$ and $7 u$ .

$u^{2} + 15 u + 8 \ln (| u |) + 14 \ln (| u |) - 3 \ln (| u |) + C$

Step 29.2.2

Add $8 \ln (| u |)$ and $14 \ln (| u |)$ .

$u^{2} + 15 u + 22 \ln (| u |) - 3 \ln (| u |) + C$

Step 29.2.3

Subtract $3 \ln (| u |)$ from $22 \ln (| u |)$ .

$u^{2} + 15 u + 19 \ln (| u |) + C$

Step 30

Replace all occurrences of $u$ with $x - 2$ .

${(x - 2)}^{2} + 15 (x - 2) + 19 \ln (| x - 2 |) + C$