Calculus Examples

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

Step 1

Let $u = x^{2} + 1$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{2} + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} + 1$ .

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

Step 1.1.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 1.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt{u - 1}}^{2}$ as $u - 1$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u - 1}$ as ${(u - 1)}^{\frac{1}{2}}$ .

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

Step 2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.3

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

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

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

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

Step 2.1.4.2

Rewrite the expression.

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

Step 2.1.5

Simplify.

$\int \frac{u - 1}{u} \cdot \frac{1}{2} d u$

Step 2.2

Multiply $\frac{u - 1}{u}$ by $\frac{1}{2}$ .

$\int \frac{u - 1}{u \cdot 2} d u$

Step 2.3

Move $2$ to the left of $u$ .

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

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 4

Divide $u - 1$ by $u$ .

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Step 4.1

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 4.2

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

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

Step 4.3

Multiply the new quotient term by the divisor.

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

Step 4.4

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 4.5

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 4.6

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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

Step 10

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Simplify.

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Step 11.2.1

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

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

Step 11.2.2

Multiply $\frac{1}{2}$ by $1$ .

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

Step 11.2.3

Combine $\frac{1}{2}$ and $\ln (| x^{2} + 1 |)$ .

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

Step 11.3

Combine the numerators over the common denominator.

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

Step 12

Reorder terms.

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