Calculus Examples

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

Step 1

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x + 1$ .

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

Step 1.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 1.1.3

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

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

Step 1.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ 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 $d u$ .

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

Step 2

Split the fraction into multiple fractions.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Simplify by multiplying through.

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

Rewrite ${(u - 1)}^{2}$ as $(u - 1) (u - 1)$ .

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Apply the distributive property.

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

Step 4.6

Reorder $u$ and $- 1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the expression.

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Step 8.1

Add $1$ and $1$ .

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

Step 8.2

Multiply $- 1$ by $- 1$ .

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

Step 9

Subtract $1 u$ from $- 1 u$ .

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

Step 10

Divide $u^{2} - 2 u + 1$ by $u$ .

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Step 10.1

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

$u$

$0$

$u^{2}$

$2 u$

$1$

Step 10.2

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

					$u$
	$u$	+	$0$		$u^{2}$	-	$2 u$	+	$1$

Step 10.3

Multiply the new quotient term by the divisor.

				$u$
$u$	+	$0$		$u^{2}$	-	$2 u$	+	$1$
			+	$u^{2}$	+	$0$

Step 10.4

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

				$u$
$u$	+	$0$		$u^{2}$	-	$2 u$	+	$1$
			-	$u^{2}$	-	$0$

Step 10.5

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

				$u$
$u$	+	$0$		$u^{2}$	-	$2 u$	+	$1$
			-	$u^{2}$	-	$0$
					-	$2 u$

Step 10.6

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

				$u$
$u$	+	$0$		$u^{2}$	-	$2 u$	+	$1$
			-	$u^{2}$	-	$0$
					-	$2 u$	+	$1$

Step 10.7

Divide the highest order term in the dividend $- 2 u$ by the highest order term in divisor $u$ .

				$u$	-	$2$
$u$	+	$0$		$u^{2}$	-	$2 u$	+	$1$
			-	$u^{2}$	-	$0$
					-	$2 u$	+	$1$

Step 10.8

Multiply the new quotient term by the divisor.

				$u$	-	$2$
$u$	+	$0$		$u^{2}$	-	$2 u$	+	$1$
			-	$u^{2}$	-	$0$
					-	$2 u$	+	$1$
					-	$2 u$	+	$0$

Step 10.9

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

				$u$	-	$2$
$u$	+	$0$		$u^{2}$	-	$2 u$	+	$1$
			-	$u^{2}$	-	$0$
					-	$2 u$	+	$1$
					+	$2 u$	-	$0$

Step 10.10

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

				$u$	-	$2$
$u$	+	$0$		$u^{2}$	-	$2 u$	+	$1$
			-	$u^{2}$	-	$0$
					-	$2 u$	+	$1$
					+	$2 u$	-	$0$
							+	$1$

Step 10.11

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

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

Step 11

Split the single integral into multiple integrals.

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

Step 12

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

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

Step 13

Apply the constant rule.

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

Step 14

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

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

Step 15

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

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

Step 16

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

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

Step 17

Multiply $3$ by $- 1$ .

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

Step 18

Divide $u - 1$ by $u$ .

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Step 18.1

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

$u$

$0$

$u$

$1$

Step 18.2

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

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

Step 18.3

Multiply the new quotient term by the divisor.

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

Step 18.4

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

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

Step 18.5

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

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

Step 18.6

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

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

Step 19

Split the single integral into multiple integrals.

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

Step 20

Apply the constant rule.

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

Step 21

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

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

Step 22

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

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

Step 23

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

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

Step 24

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

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

Step 25

Simplify.

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

Step 26

Reorder terms.

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

Step 27

Simplify.

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Step 27.1

Subtract $3 u$ from $- 2 u$ .

$\frac{1}{2} u^{2} - 5 u + \ln (| u |) + 3 \ln (| u |) + 2 \ln (| u |) + C$

Step 27.2

Add $\ln (| u |)$ and $3 \ln (| u |)$ .

$\frac{1}{2} u^{2} - 5 u + 4 \ln (| u |) + 2 \ln (| u |) + C$

Step 27.3

Add $4 \ln (| u |)$ and $2 \ln (| u |)$ .

$\frac{1}{2} u^{2} - 5 u + 6 \ln (| u |) + C$

Step 28

Replace all occurrences of $u$ with $x + 1$ .

$\frac{1}{2} {(x + 1)}^{2} - 5 (x + 1) + 6 \ln (| x + 1 |) + C$