Calculus Examples

$\frac{7 x^{4} + 5}{x^{2} + 1}$

Step 1

Write $\frac{7 x^{4} + 5}{x^{2} + 1}$ as a function.

$f (x) = \frac{7 x^{4} + 5}{x^{2} + 1}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{7 x^{4} + 5}{x^{2} + 1} d x$

Step 4

Divide $7 x^{4} + 5$ by $x^{2} + 1$ .

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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 $0$ .

$x^{2}$

$0 x$

$1$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

Step 4.2

Divide the highest order term in the dividend $7 x^{4}$ by the highest order term in divisor $x^{2}$ .

$7 x^{2}$

$x^{2}$

$0 x$

$1$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

Step 4.3

Multiply the new quotient term by the divisor.

$7 x^{2}$

$x^{2}$

$0 x$

$1$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$7 x^{4}$

$0$

$7 x^{2}$

Step 4.4

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{4} + 0 + 7 x^{2}$

$7 x^{2}$

$x^{2}$

$0 x$

$1$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$7 x^{4}$

$0$

$7 x^{2}$

Step 4.5

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

$7 x^{2}$

$x^{2}$

$0 x$

$1$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$7 x^{4}$

$0$

$7 x^{2}$

Step 4.6

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

$7 x^{2}$

$x^{2}$

$0 x$

$1$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$7 x^{4}$

$0$

$7 x^{2}$

$0 x$

$5$

Step 4.7

Divide the highest order term in the dividend $- 7 x^{2}$ by the highest order term in divisor $x^{2}$ .

$7 x^{2}$

$0 x$

$7$

$x^{2}$

$0 x$

$1$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$7 x^{4}$

$0$

$7 x^{2}$

$0 x$

$5$

Step 4.8

Multiply the new quotient term by the divisor.

$7 x^{2}$

$0 x$

$7$

$x^{2}$

$0 x$

$1$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$7 x^{4}$

$0$

$7 x^{2}$

$0 x$

$5$

$7 x^{2}$

$0$

$7$

Step 4.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{2} + 0 - 7$

$7 x^{2}$

$0 x$

$7$

$x^{2}$

$0 x$

$1$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$7 x^{4}$

$0$

$7 x^{2}$

$0 x$

$5$

$7 x^{2}$

$0$

$7$

Step 4.10

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

$7 x^{2}$

$0 x$

$7$

$x^{2}$

$0 x$

$1$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$7 x^{4}$

$0$

$7 x^{2}$

$0 x$

$5$

$7 x^{2}$

$0$

$7$

$12$

Step 4.11

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

$\int 7 x^{2} - 7 + \frac{12}{x^{2} + 1} d x$

Step 5

Split the single integral into multiple integrals.

$\int 7 x^{2} d x + \int - 7 d x + \int \frac{12}{x^{2} + 1} d x$

Step 6

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int x^{2} d x + \int - 7 d x + \int \frac{12}{x^{2} + 1} d x$

Step 7

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

$7 (\frac{1}{3} x^{3} + C) + \int - 7 d x + \int \frac{12}{x^{2} + 1} d x$

Step 8

Apply the constant rule.

$7 (\frac{1}{3} x^{3} + C) - 7 x + C + \int \frac{12}{x^{2} + 1} d x$

Step 9

Combine $\frac{1}{3}$ and $x^{3}$ .

$7 (\frac{x^{3}}{3} + C) - 7 x + C + \int \frac{12}{x^{2} + 1} d x$

Step 10

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$7 (\frac{x^{3}}{3} + C) - 7 x + C + 12 \int \frac{1}{x^{2} + 1} d x$

Step 11

Simplify the expression.

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Step 11.1

Reorder $x^{2}$ and $1$ .

$7 (\frac{x^{3}}{3} + C) - 7 x + C + 12 \int \frac{1}{1 + x^{2}} d x$

Step 11.2

Rewrite $1$ as $1^{2}$ .

$7 (\frac{x^{3}}{3} + C) - 7 x + C + 12 \int \frac{1}{1^{2} + x^{2}} d x$

Step 12

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

$7 (\frac{x^{3}}{3} + C) - 7 x + C + 12 (\arctan (x) + C)$

Step 13

Simplify.

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Step 13.1

Simplify.

$\frac{7 x^{3}}{3} - 7 x + 12 \arctan (x) + C$

Step 13.2

Reorder terms.

$\frac{7}{3} x^{3} - 7 x + 12 \arctan (x) + C$

Step 14

The answer is the antiderivative of the function $f (x) = \frac{7 x^{4} + 5}{x^{2} + 1}$ .

$F (x) =$ $\frac{7}{3} x^{3} - 7 x + 12 \arctan (x) + C$