Respuesta :

Space

Answer:

[tex]\displaystyle \int {(hx + q)} \, dx = \frac{hx^2}{2} + qx + C[/tex]

General Formulas and Concepts:

Calculus

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]

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]

Step-by-step explanation:

*Note:

Assume h and q are arbitrary constants, where h and q ≠ 0.

Step 1: Define

Identify

[tex]\displaystyle \int {(hx + q)} \, dx[/tex]

Step 2: Integrate

  1. [Integral] Rewrite [Integration Property - Addition/Subtraction]:               [tex]\displaystyle \int {(hx + q)} \, dx = \int {hx} \, dx + \int {q} \, dx[/tex]
  2. [Integrals] Rewrite [Integration Property - Multiplied Constant]:               [tex]\displaystyle \int {(hx + q)} \, dx = h\int {x} \, dx + q\int {} \, dx[/tex]
  3. [Integrals] Reverse Power Rule:                                                                   [tex]\displaystyle \int {(hx + q)} \, dx = h \Big( \frac{x^2}{2} \Big) + qx + C[/tex]
  4. Simplify:                                                                                                         [tex]\displaystyle \int {(hx + q)} \, dx = \frac{hx^2}{2} + qx + C[/tex]

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

Unit:  Integration

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