annesosaindon
Answered

integrating secx(sex+tanx) and the steps please!

i'm not sure which technique is supposed to be used here.. ex; integration by parts or substitution etc. Thanks !

Verified Answer :

TheHero

Simply distrbution and direct application of memorized integrals


I will refer the integration sign as int

J = Int secx(secx + tanx ) dx = int sec^2x + secxtanx dx

Remember int sec^2x dx= tanx +c
Int secxtanx = secx +c

If you wanna their proof just request

J = tanx + secx +c

Space

Answer:

\(\displaystyle \int {\sec x(\sec x + \tan x)} \, dx = \tan x + \sec x + C\)

General Formulas and Concepts:

Calculus

Integration

  • Integrals
  • [Indefinite Integrals] Integration Constant C

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

Step-by-step explanation:

Step 1: Define

Identify

\(\displaystyle \int {\sec x(\sec x + \tan x)} \, dx\)

Step 2: Integrate

  1. [Integrand] Rewrite:                                                                                          \(\displaystyle \int {\sec x(\sec x + \tan x)} \, dx = \int {\sec^2x + \sec x \tan x} \, dx\)
  2. [Integral] Rewrite [Integration Property - Addition/Subtraction]:                   \(\displaystyle \int {\sec x(\sec x + \tan x)} \, dx = \int {\sec^2x} \, dx + \int {\sec x \tan x} \, dx\)
  3. [Integrals] Trigonometric Integration:                                                               \(\displaystyle \int {\sec x(\sec x + \tan x)} \, dx = \tan x + \sec x + C\)

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

Unit:  Integration

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