Calculus Examples

$(x + 1) (x + 2) (x + 3)$

Step 1

Write $(x + 1) (x + 2) (x + 3)$ as a function.

$f (x) = (x + 1) (x + 2) (x + 3)$

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.

$F (x) = \int (x + 1) (x + 2) (x + 3) d x$

Step 4

Let $u = x + 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.1

Let $u = x + 3$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $x + 3$ .

$\frac{d}{d x} [x + 3]$

Step 4.1.2

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 4.1.3

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

$1 + \frac{d}{d x} [3]$

Step 4.1.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$1 + 0$

Step 4.1.5

Add $1$ and $0$ .

$1$

Step 4.2

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

$\int (u - 3 + 1) (u - 3 + 2) u d u$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Add $- 3$ and $1$ .

$\int (u - 2) (u - 3 + 2) u d u$

Step 5.2

Add $- 3$ and $2$ .

$\int (u - 2) (u - 1) u d u$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

$\int (u (u - 1) - 2 (u - 1)) u d u$

Step 6.2

Apply the distributive property.

$\int (u \cdot u + u \cdot - 1 - 2 (u - 1)) u d u$

Step 6.3

Apply the distributive property.

$\int (u \cdot u + u \cdot - 1 - 2 u - 2 \cdot - 1) u d u$

Step 6.4

Apply the distributive property.

$\int (u \cdot u + u \cdot - 1) u + (- 2 u - 2 \cdot - 1) u d u$

Step 6.5

Apply the distributive property.

$\int u \cdot u \cdot u + u \cdot - 1 u + (- 2 u - 2 \cdot - 1) u d u$

Step 6.6

Apply the distributive property.

$\int u \cdot u \cdot u + u \cdot - 1 u - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.7

Reorder $u$ and $- 1$ .

$\int u \cdot u \cdot u - 1 \cdot u \cdot u - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.8

Raise $u$ to the power of $1$ .

$\int u^{1} u \cdot u - 1 u \cdot u - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.9

Raise $u$ to the power of $1$ .

$\int u^{1} u^{1} u - 1 u \cdot u - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u^{1 + 1} u - 1 u \cdot u - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.11

Add $1$ and $1$ .

$\int u^{2} u - 1 u \cdot u - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.12

Raise $u$ to the power of $1$ .

$\int u^{2} u^{1} - 1 u \cdot u - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.13

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u^{2 + 1} - 1 u \cdot u - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.14

Add $2$ and $1$ .

$\int u^{3} - 1 u \cdot u - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.15

Factor out negative.

$\int u^{3} - (u \cdot u) - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.16

Raise $u$ to the power of $1$ .

$\int u^{3} - (u^{1} u) - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.17

Raise $u$ to the power of $1$ .

$\int u^{3} - (u^{1} u^{1}) - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.18

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u^{3} - u^{1 + 1} - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.19

Add $1$ and $1$ .

$\int u^{3} - u^{2} - 2 u \cdot u - 2 \cdot - 1 u d u$

Step 6.20

Raise $u$ to the power of $1$ .

$\int u^{3} - u^{2} - 2 (u^{1} u) - 2 \cdot - 1 u d u$

Step 6.21

Raise $u$ to the power of $1$ .

$\int u^{3} - u^{2} - 2 (u^{1} u^{1}) - 2 \cdot - 1 u d u$

Step 6.22

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u^{3} - u^{2} - 2 u^{1 + 1} - 2 \cdot - 1 u d u$

Step 6.23

Add $1$ and $1$ .

$\int u^{3} - u^{2} - 2 u^{2} - 2 \cdot - 1 u d u$

Step 6.24

Multiply $- 2$ by $- 1$ .

$\int u^{3} - u^{2} - 2 u^{2} + 2 u d u$

Step 6.25

Subtract $2 u^{2}$ from $- u^{2}$ .

$\int u^{3} - 3 u^{2} + 2 u d u$

Step 7

Split the single integral into multiple integrals.

$\int u^{3} d u + \int - 3 u^{2} d u + \int 2 u d u$

Step 8

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

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

Step 9

Since $- 3$ is constant with respect to $u$ , move $- 3$ out of the integral.

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

Step 10

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

$\frac{1}{4} u^{4} + C - 3 (\frac{1}{3} u^{3} + C) + \int 2 u d u$

Step 11

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

$\frac{1}{4} u^{4} + C - 3 (\frac{1}{3} u^{3} + C) + 2 \int u d u$

Step 12

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

$\frac{1}{4} u^{4} + C - 3 (\frac{1}{3} u^{3} + C) + 2 (\frac{1}{2} u^{2} + C)$

Step 13

Simplify.

Tap for more steps...

Step 13.1

Simplify.

$\frac{u^{4}}{4} - u^{3} + 2 (\frac{1}{2} u^{2}) + C$

Step 13.2

Simplify.

Tap for more steps...

Step 13.2.1

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

$\frac{u^{4}}{4} - u^{3} + 2 \frac{u^{2}}{2} + C$

Step 13.2.2

Combine $2$ and $\frac{u^{2}}{2}$ .

$\frac{u^{4}}{4} - u^{3} + \frac{2 u^{2}}{2} + C$

Step 13.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 13.2.3.1

Cancel the common factor.

$\frac{u^{4}}{4} - u^{3} + \frac{2 u^{2}}{2} + C$

Step 13.2.3.2

Divide $u^{2}$ by $1$ .

$\frac{u^{4}}{4} - u^{3} + u^{2} + C$

Step 14

Replace all occurrences of $u$ with $x + 3$ .

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

Step 15

Reorder terms.

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

Step 16

The answer is the antiderivative of the function $f (x) = (x + 1) (x + 2) (x + 3)$ .

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