Calculus Examples

$\int \frac{2 x^{3}}{x^{2} - 4} d x$

Step 1

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

$2 \int \frac{x^{3}}{x^{2} - 4} d x$

Step 2

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

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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 $0$ .

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 2.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x^{2}$ .

$x$

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 2.3

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$4 x$

Step 2.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 0 - 4 x$

$x$

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$4 x$

Step 2.5

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

$x$

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$4 x$

Step 2.6

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

$x$

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$4 x$

$0$

Step 2.7

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

$2 \int x + \frac{4 x}{x^{2} - 4} d x$

Step 3

Split the single integral into multiple integrals.

$2 (\int x d x + \int \frac{4 x}{x^{2} - 4} d x)$

Step 4

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

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

Step 5

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

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

Step 6

Let $u = x^{2} - 4$ . 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 6.1

Let $u = x^{2} - 4$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $x^{2} - 4$ .

$\frac{d}{d x} [x^{2} - 4]$

Step 6.1.2

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$

Step 6.1.3

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

$2 x + \frac{d}{d x} [- 4]$

Step 6.1.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$2 x + 0$

Step 6.1.5

Add $2 x$ and $0$ .

$2 x$

Step 6.2

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

$2 (\frac{1}{2} x^{2} + C + 4 \int \frac{1}{u} \cdot \frac{1}{2} d u)$

Step 7

Simplify.

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Step 7.1

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

$2 (\frac{1}{2} x^{2} + C + 4 \int \frac{1}{u \cdot 2} d u)$

Step 7.2

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

$2 (\frac{1}{2} x^{2} + C + 4 \int \frac{1}{2 \cdot u} d u)$

Step 7.3

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{2}$ and $4$ .

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

Step 9.2

Cancel the common factor of $4$ and $2$ .

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Step 9.2.1

Factor $2$ out of $4$ .

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

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $2$ out of $2$ .

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

Step 9.2.2.2

Cancel the common factor.

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

Step 9.2.2.3

Rewrite the expression.

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

Step 9.2.2.4

Divide $2$ by $1$ .

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

Step 10

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

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

Step 11

Simplify.

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

Step 12

Replace all occurrences of $u$ with $x^{2} - 4$ .

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

Step 13

Simplify.

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Step 13.1

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

$2 (\frac{x^{2}}{2} + 2 \ln (| x^{2} - 4 |)) + C$

Step 13.2

To write $2 \ln (| x^{2} - 4 |)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 (\frac{x^{2}}{2} + 2 \ln (| x^{2} - 4 |) \cdot \frac{2}{2}) + C$

Step 13.3

Combine $2 \ln (| x^{2} - 4 |)$ and $\frac{2}{2}$ .

$2 (\frac{x^{2}}{2} + \frac{2 \ln (| x^{2} - 4 |) \cdot 2}{2}) + C$

Step 13.4

Combine the numerators over the common denominator.

$2 \frac{x^{2} + 2 \ln (| x^{2} - 4 |) \cdot 2}{2} + C$

Step 13.5

Cancel the common factor of $2$ .

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Step 13.5.1

Cancel the common factor.

$2 \frac{x^{2} + 2 \ln (| x^{2} - 4 |) \cdot 2}{2} + C$

Step 13.5.2

Rewrite the expression.

$x^{2} + 2 \ln (| x^{2} - 4 |) \cdot 2 + C$

Step 13.6

Multiply $2$ by $2$ .

$x^{2} + 4 \ln (| x^{2} - 4 |) + C$