Definite Integration
Substitution method
Grade Class 12
Question:
The integral ∫ sec^2 x / (sec x + tan x)^(9/2) dx equals (for some arbitrary constant K)
(A) - 1 / (sec x + tan x)^(11/2) * {1/11 - 1/7 * (sec x + tan x)^2} + K
(B) - 1 / (sec x + tan x)^(11/2) * {1/11 - 1/7 * (sec x + tan x)^2} + K
(C) - 1 / (sec x + tan x)^(11/2) * {1/11 + 1/7 * (sec x + tan x)^2} + K
(D) - 1 / (sec x + tan x)^(11/2) * {1/11 + 1/7 * (sec x + tan x)^2} + K
Step-by-Step Solution
Key Concept: Let sec x + tan x = t, then sec x(sec x + tan x) dx = dt
Let sec x + tan x = t. Then sec x(sec x + tan x) dx = dt. So sec x dx = dt/t. Also sec x - tan x = 1/t. Adding gives 2 sec x = t + 1/t, so sec x = (t^2 + 1)/2t. Integral becomes \int (t^2 + 1)/2t * (1/t) / t^(9/2) dt = 1/2 \int (t^2 + 1) / t^(13/2) dt.
Correct Answer: C