Question

Integrate the following•(x+10)²​

Answer 1

Answer:

$\rm\displaystyle \frac{ {x}^{3} }{3} + 10 {x}^{2} + 100 x + \rm C$

Step-by-step 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

Answer:

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

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

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]:                                                               $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

U-Substitution

Step-by-step 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:                                                                     $\displaystyle \int {(x + 10)^2} \, dx = \frac{u^3}{3} + C$
3. Back-Substitute:                                                                                             $\displaystyle \int {(x + 10)^2} \, dx = \frac{(x + 10)^3}{3} + C$

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

Unit: Integration

Book: College Calculus 10e