Respuesta :

Space

Answer:

[tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = 2 \arctan x - \frac{5}{2(x^2 + 1)} + C[/tex]

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                       [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

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

Derivative Rule [Basic Power Rule]:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals

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

Integration Property [Multiplied Constant]:                                                     [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Property [Addition/Subtraction]:                                                   [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Integration Method: U-Substitution

Step-by-step explanation:

Step 1: Define

Identify.

[tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx[/tex]

Step 2: Integrate Pt. 1

  1. [Integrand] Rewrite [Factor]:                                                                     [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = \int {\frac{2(x^2 + 1) + 5x}{(x^2 + 1)^2}} \, dx[/tex]
  2. Rewrite [Integration Property - Addition/Subtraction]:                           [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = \int {\frac{2(x^2 + 1)}{(x^2 + 1)^2}} \, dx + \int {\frac{5x}{(x^2 + 1)^2}} \, dx[/tex]
  3. [Left Integral] Simplify:                                                                               [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = \int {\frac{2}{x^2 + 1}} \, dx + \int {\frac{5x}{(x^2 + 1)^2}} \, dx[/tex]
  4. [Integrals] Rewrite [Integration Property - Multiplied Constant]:           [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = 2 \int {\frac{1}{x^2 + 1}} \, dx + 5 \int {\frac{x}{(x^2 + 1)^2}} \, dx[/tex]
  5. [Left Integral] Trigonometric Integration:                                                 [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = 2 \arctan x + 5 \int {\frac{x}{(x^2 + 1)^2}} \, dx[/tex]

Step 3: Integrate Pt. 2

Identify variables for u-substitution for the remaining integral.

  1. Set u:                                                                                                         [tex]\displaystyle u = x^2 + 1[/tex]
  2. [u] Differentiate [Derivative Properties and Rules]:                                 [tex]\displaystyle du = 2x \ dx[/tex]

Step 4: Integrate Pt. 3

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:             [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = 2 \arctan x + \frac{5}{2} \int {\frac{2x}{(x^2 + 1)^2}} \, dx[/tex]
  2. [Integral] U-Substitution:                                                                           [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = 2 \arctan x + \frac{5}{2} \int {\frac{1}{u^2}} \, du[/tex]
  3. Apply Integration Rule [Reverse Power Rule]:                                       [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = 2 \arctan x + \frac{5}{2} \bigg( \frac{-1}{u} \bigg) + C[/tex]
  4. [u] Back-substitute:                                                                                   [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = 2 \arctan x + \frac{5}{2} \bigg( \frac{-1}{x^2 + 1} \bigg) + C[/tex]
  5. Simplify:                                                                                                     [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx = 2 \arctan x - \frac{5}{2(x^2 + 1)} + C[/tex]

∴ the integration of the given integral   [tex]\displaystyle \int {\frac{2x^2 + 5x + 2}{(x^2 + 1)^2}} \, dx[/tex]  is equal to  [tex]\displaystyle \bold{2 \arctan x - \frac{5}{2(x^2 + 1)} + C}[/tex].

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Learn more about integration: https://brainly.com/question/21023287

Learn more about calculus: https://brainly.com/question/23558817

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

Unit: Integration

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