Question

Integrate the following
•(x+10)²​

Answer 1

`\rm\displaystyle \frac{ {x}^(3) }{3} + 10 {x}^(2) + 100 x + \rm C`

Explanation:

we would like to integrate the following integration:

`\displaystyle \int (x + 10 {)}^(2) dx`

to do so simplify the integrand by using algebraic identity which yields:

`\displaystyle \int {x}^(2) + 20x + 100 dx`

by sum integration we obtain:

`\rm\displaystyle \int {x}^(2)dx + \int20x dx+ \int 100 dx`

remember that,

`\displaystyle \int cxdx = c \int xdx`

so,we acquire:

`\rm\displaystyle \int {x}^(2)dx + 20\int x dx+ \int 100 dx`

use exponent integration rule which yields:

`\rm\displaystyle \frac{ {x}^(3) }{3} + 20 \frac{ {x}^(2) }{2} + \int 100 dx`

use constant integration rule which yields:

`\rm\displaystyle \frac{ {x}^(3) }{3} + 20 \frac{ {x}^(2) }{2} + 100 x`

simplify:

`\rm\displaystyle \frac{ {x}^(3) }{3} + 10 {x}^(2) + 100 x`

and we of course have to add the constant of integration:

`\rm\displaystyle \frac{ {x}^(3) }{3} + 10 {x}^(2) + 100 x + \rm C`

Answer 2

`\displaystyle \int {(x + 10)^2} \, dx = ((x + 10)^3)/(3) + C`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals
• Indefinite Integrals
• Integration Constant C

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (x^(n + 1))/(n + 1) + C`

U-Substitution

Explanation:

*Note:

The answer below me is correct, but there is a simpler method to obtain the answer.

Step 1: Define

Identify

`\displaystyle \int {(x + 10)^2} \, dx`

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

1. Set u:
   `\displaystyle u = x + 10`
2. [u] Differentiate [Basic Power Rule]:
   `\displaystyle du = dx`

Step 3: Integrate Pt. 2

1. [Integral] U-Substitution:
   `\displaystyle \int {(x + 10)^2} \, dx = \int {u^2} \, du`
2. [Integral] Reverse Power Rule:
   `\displaystyle \int {(x + 10)^2} \, dx = (u^3)/(3) + C`
3. Back-Substitute:
   `\displaystyle \int {(x + 10)^2} \, dx = ((x + 10)^3)/(3) + C`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Book: College Calculus 10e