Question

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 !

Answer 1

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

Answer 2

`\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`

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