Calculus Examples

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

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 $d u$ .

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

Step 2

Simplify by multiplying through.

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Step 2.1

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

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 2.4

Apply the distributive property.

$\int \frac{u \cdot u + u \cdot 1 + 1 u + 1 \cdot 1}{u} d u$

Step 2.5

Reorder $u$ and $1$ .

$\int \frac{u \cdot u + 1 \cdot u + 1 u + 1 \cdot 1}{u} d u$

Step 3

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

$\int \frac{u^{1} u + 1 u + 1 u + 1 \cdot 1}{u} d u$

Step 4

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

$\int \frac{u^{1} u^{1} + 1 u + 1 u + 1 \cdot 1}{u} d u$

Step 5

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

$\int \frac{u^{1 + 1} + 1 u + 1 u + 1 \cdot 1}{u} d u$

Step 6

Simplify the expression.

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Step 6.1

Add $1$ and $1$ .

$\int \frac{u^{2} + 1 u + 1 u + 1 \cdot 1}{u} d u$

Step 6.2

Multiply $u$ by $1$ .

$\int \frac{u^{2} + u + 1 u + 1 \cdot 1}{u} d u$

Step 6.3

Multiply $u$ by $1$ .

$\int \frac{u^{2} + u + u + 1 \cdot 1}{u} d u$

Step 6.4

Multiply $1$ by $1$ .

$\int \frac{u^{2} + u + u + 1}{u} d u$

Step 7

Add $u$ and $u$ .

$\int \frac{u^{2} + 2 u + 1}{u} d u$

Step 8

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

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Step 8.1

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 8.2

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 8.3

Multiply the new quotient term by the divisor.

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

Step 8.4

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 8.5

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 8.6

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 8.7

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 8.8

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 8.9

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 8.10

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 8.11

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

$\int u + 2 + \frac{1}{u} d u$

Step 9

Split the single integral into multiple integrals.

$\int u d u + \int 2 d u + \int \frac{1}{u} d u$

Step 10

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

$\frac{1}{2} u^{2} + C + \int 2 d u + \int \frac{1}{u} d u$

Step 11

Apply the constant rule.

$\frac{1}{2} u^{2} + C + 2 u + C + \int \frac{1}{u} d u$

Step 12

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

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

Step 13

Simplify.

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

Step 14

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

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