Calculus Examples

$\int \frac{x^{3}}{100 + x^{2}} d x$

Step 1

Reorder $100$ and $x^{2}$ .

$\int \frac{x^{3}}{x^{2} + 100} d x$

Step 2

Divide $x^{3}$ by $x^{2} + 100$ .

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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 $0$ .

$x^{2}$

$0 x$

$100$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 2.2

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

$x$

$x^{2}$

$0 x$

$100$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 2.3

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$0 x$

$100$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$100 x$

Step 2.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 0 + 100 x$

$x$

$x^{2}$

$0 x$

$100$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$100 x$

Step 2.5

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

$x$

$x^{2}$

$0 x$

$100$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$100 x$

Step 2.6

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

$x$

$x^{2}$

$0 x$

$100$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$100 x$

$0$

Step 2.7

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

$\int x + \frac{- 100 x}{x^{2} + 100} d x$

Step 3

Split the single integral into multiple integrals.

$\int x d x + \int \frac{- 100 x}{x^{2} + 100} d x$

Step 4

Move the negative in front of the fraction.

$\int x d x + \int - \frac{100 x}{x^{2} + 100} d x$

Step 5

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

$\frac{1}{2} x^{2} + C + \int - \frac{100 x}{x^{2} + 100} d x$

Step 6

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

$\frac{1}{2} x^{2} + C - \int \frac{100 x}{x^{2} + 100} d x$

Step 7

Since $100$ is constant with respect to $x$ , move $100$ out of the integral.

$\frac{1}{2} x^{2} + C - (100 \int \frac{x}{x^{2} + 100} d x)$

Step 8

Multiply $100$ by $- 1$ .

$\frac{1}{2} x^{2} + C - 100 \int \frac{x}{x^{2} + 100} d x$

Step 9

Let $u = x^{2} + 100$ . 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 9.1

Let $u = x^{2} + 100$ . Find $\frac{d u}{d x}$ .

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Step 9.1.1

Differentiate $x^{2} + 100$ .

$\frac{d}{d x} [x^{2} + 100]$

Step 9.1.2

By the Sum Rule, the derivative of $x^{2} + 100$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [100]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [100]$

Step 9.1.3

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

$2 x + \frac{d}{d x} [100]$

Step 9.1.4

Since $100$ is constant with respect to $x$ , the derivative of $100$ with respect to $x$ is $0$ .

$2 x + 0$

Step 9.1.5

Add $2 x$ and $0$ .

$2 x$

Step 9.2

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

$\frac{1}{2} x^{2} + C - 100 \int \frac{1}{u} \cdot \frac{1}{2} d u$

Step 10

Simplify.

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Step 10.1

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

$\frac{1}{2} x^{2} + C - 100 \int \frac{1}{u \cdot 2} d u$

Step 10.2

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

$\frac{1}{2} x^{2} + C - 100 \int \frac{1}{2 u} d u$

Step 11

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

$\frac{1}{2} x^{2} + C - 100 (\frac{1}{2} \int \frac{1}{u} d u)$

Step 12

Simplify.

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Step 12.1

Combine $\frac{1}{2}$ and $- 100$ .

$\frac{1}{2} x^{2} + C + \frac{- 100}{2} \int \frac{1}{u} d u$

Step 12.2

Cancel the common factor of $- 100$ and $2$ .

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Step 12.2.1

Factor $2$ out of $- 100$ .

$\frac{1}{2} x^{2} + C + \frac{2 \cdot - 50}{2} \int \frac{1}{u} d u$

Step 12.2.2

Cancel the common factors.

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Step 12.2.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} x^{2} + C + \frac{2 \cdot - 50}{2 (1)} \int \frac{1}{u} d u$

Step 12.2.2.2

Cancel the common factor.

$\frac{1}{2} x^{2} + C + \frac{2 \cdot - 50}{2 \cdot 1} \int \frac{1}{u} d u$

Step 12.2.2.3

Rewrite the expression.

$\frac{1}{2} x^{2} + C + \frac{- 50}{1} \int \frac{1}{u} d u$

Step 12.2.2.4

Divide $- 50$ by $1$ .

$\frac{1}{2} x^{2} + C - 50 \int \frac{1}{u} d u$

Step 13

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

$\frac{1}{2} x^{2} + C - 50 (\ln (| u |) + C)$

Step 14

Simplify.

$\frac{1}{2} x^{2} - 50 \ln (| u |) + C$

Step 15

Replace all occurrences of $u$ with $x^{2} + 100$ .

$\frac{1}{2} x^{2} - 50 \ln (| x^{2} + 100 |) + C$