Calculus Examples

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

Step 1

Divide $x^{4}$ by $x^{2} + 4$ .

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Step 1.1

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

$x^{2}$

$0 x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 1.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$0 x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 1.3

Multiply the new quotient term by the divisor.

$x^{2}$

$0 x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$4 x^{2}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 0 + 4 x^{2}$

$x^{2}$

$0 x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$4 x^{2}$

Step 1.5

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

$x^{2}$

$0 x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$4 x^{2}$

Step 1.6

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

$x^{2}$

$0 x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$4 x^{2}$

$0 x$

$0$

Step 1.7

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$0 x$

$4$

$x^{2}$

$0 x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$4 x^{2}$

$0 x$

$0$

Step 1.8

Multiply the new quotient term by the divisor.

$x^{2}$

$0 x$

$4$

$x^{2}$

$0 x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$4 x^{2}$

$0 x$

$0$

$4 x^{2}$

$0$

$16$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 0 - 16$

$x^{2}$

$0 x$

$4$

$x^{2}$

$0 x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$4 x^{2}$

$0 x$

$0$

$4 x^{2}$

$0$

$16$

Step 1.10

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

$x^{2}$

$0 x$

$4$

$x^{2}$

$0 x$

$4$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0$

$4 x^{2}$

$0 x$

$0$

$4 x^{2}$

$0$

$16$

Step 1.11

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

$\int x^{2} - 4 + \frac{16}{x^{2} + 4} d x$

Step 2

Split the single integral into multiple integrals.

$\int x^{2} d x + \int - 4 d x + \int \frac{16}{x^{2} + 4} d x$

Step 3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3} + C + \int - 4 d x + \int \frac{16}{x^{2} + 4} d x$

Step 4

Apply the constant rule.

$\frac{1}{3} x^{3} + C - 4 x + C + \int \frac{16}{x^{2} + 4} d x$

Step 5

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$\frac{1}{3} x^{3} + C - 4 x + C + 16 \int \frac{1}{x^{2} + 4} d x$

Step 6

Simplify the expression.

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Step 6.1

Reorder $x^{2}$ and $4$ .

$\frac{1}{3} x^{3} + C - 4 x + C + 16 \int \frac{1}{4 + x^{2}} d x$

Step 6.2

Rewrite $4$ as $2^{2}$ .

$\frac{1}{3} x^{3} + C - 4 x + C + 16 \int \frac{1}{2^{2} + x^{2}} d x$

Step 7

The integral of $\frac{1}{2^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{2} \arctan (\frac{x}{2}) + C$ .

$\frac{1}{3} x^{3} + C - 4 x + C + 16 (\frac{1}{2} \arctan (\frac{x}{2}) + C)$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{2}$ and $\arctan (\frac{x}{2})$ .

$\frac{1}{3} x^{3} + C - 4 x + C + 16 (\frac{\arctan (\frac{x}{2})}{2} + C)$

Step 8.2

Simplify.

$\frac{1}{3} x^{3} - 4 x + 16 (\frac{1}{2}) \arctan (\frac{1}{2} x) + C$

Step 8.3

Simplify.

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Step 8.3.1

Combine $16$ and $\frac{1}{2}$ .

$\frac{1}{3} x^{3} - 4 x + \frac{16}{2} \arctan (\frac{1}{2} x) + C$

Step 8.3.2

Cancel the common factor of $16$ and $2$ .

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Step 8.3.2.1

Factor $2$ out of $16$ .

$\frac{1}{3} x^{3} - 4 x + \frac{2 \cdot 8}{2} \arctan (\frac{1}{2} x) + C$

Step 8.3.2.2

Cancel the common factors.

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Step 8.3.2.2.1

Factor $2$ out of $2$ .

$\frac{1}{3} x^{3} - 4 x + \frac{2 \cdot 8}{2 (1)} \arctan (\frac{1}{2} x) + C$

Step 8.3.2.2.2

Cancel the common factor.

$\frac{1}{3} x^{3} - 4 x + \frac{2 \cdot 8}{2 \cdot 1} \arctan (\frac{1}{2} x) + C$

Step 8.3.2.2.3

Rewrite the expression.

$\frac{1}{3} x^{3} - 4 x + \frac{8}{1} \arctan (\frac{1}{2} x) + C$

Step 8.3.2.2.4

Divide $8$ by $1$ .

$\frac{1}{3} x^{3} - 4 x + 8 \arctan (\frac{1}{2} x) + C$

Step 8.3.3

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

$\frac{1}{3} x^{3} - 4 x + 8 \arctan (\frac{x}{2}) + C$

Step 8.4

Reorder terms.

$\frac{1}{3} x^{3} - 4 x + 8 \arctan (\frac{1}{2} x) + C$