Respuesta :

Nayefx

Answer:

[tex] \rm\displaystyle \frac{ {x}^{3} }{3} + 10 {x}^{2} + 100 x + \rm C[/tex]

Step-by-step explanation:

we would like to integrate the following integration:

[tex] \displaystyle \int (x + 10 {)}^{2} dx[/tex]

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

[tex] \displaystyle \int {x}^{2} + 20x + 100 dx[/tex]

by sum integration we obtain:

[tex] \rm\displaystyle \int {x}^{2}dx + \int20x dx+ \int 100 dx[/tex]

remember that,

[tex] \displaystyle \int cxdx = c \int xdx[/tex]

so,we acquire:

[tex] \rm\displaystyle \int {x}^{2}dx + 20\int x dx+ \int 100 dx[/tex]

use exponent integration rule which yields:

[tex] \rm\displaystyle \frac{ {x}^{3} }{3} + 20 \frac{ {x}^{2} }{2} + \int 100 dx[/tex]

use constant integration rule which yields:

[tex] \rm\displaystyle \frac{ {x}^{3} }{3} + 20 \frac{ {x}^{2} }{2} + 100 x[/tex]

simplify:

[tex] \rm\displaystyle \frac{ {x}^{3} }{3} + 10 {x}^{2} + 100 x[/tex]

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

[tex] \rm\displaystyle \frac{ {x}^{3} }{3} + 10 {x}^{2} + 100 x + \rm C[/tex]

Space

Answer:

[tex]\displaystyle \int {(x + 10)^2} \, dx = \frac{(x + 10)^3}{3} + C[/tex]

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

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

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

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

[tex]\displaystyle \int {(x + 10)^2} \, dx[/tex]

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

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

Step 3: Integrate Pt. 2

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

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

Unit: Integration

Book: College Calculus 10e

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