Calculus Examples

$\int \frac{x^{4} - 1}{x - 1} d x$

Step 1

Divide $x^{4} - 1$ by $x - 1$ .

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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 $0$ .

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 1.2

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 1.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

Step 1.4

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

Step 1.5

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

Step 1.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 1.7

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 1.8

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - x^{2}$

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 1.10

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 1.11

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 1.12

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

Step 1.13

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 1.14

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 1.15

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

Step 1.16

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

Step 1.17

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

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

Step 1.18

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

Step 1.19

The expression needs to be subtracted from the dividend, so change all the signs in $x - 1$

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

Step 1.20

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

$x^{3}$

$x^{2}$

$x$

$1$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$0 x$

$x^{2}$

$x$

$1$

$x$

$1$

$0$

Step 1.21

Since the remander is $0$ , the final answer is the quotient.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Apply the constant rule.

$\frac{1}{4} x^{4} + C + \frac{1}{3} x^{3} + C + \frac{1}{2} x^{2} + C + x + C$

Step 7

Simplify.

$\frac{1}{4} x^{4} + \frac{1}{3} x^{3} + \frac{1}{2} x^{2} + x + C$