Definite Integration
Definite Integration
nta_pyq_2025_jan
Grade 12
Question:
Let for f (x) = 7 tan x + 7 tan x - 3 tan x - 3 tan x, I = \int 8 6 4 2 1 0 \pi/4 f (x)dx and I = \int 2 \pi/4 0 xf (x)dx . Then 7I1 + 12I2 is equal to :
Step-by-Step Solution
Key Concept: Apply the core result for definite integral properties and simplify using the given constraints.
8 6 4 2 f (x) = 7 tan x + 7 tan x - 3 tan x - 3 tan x 6 2 2 2 (2) = 7 tan x (1 + tan x) - 3 tan x (1 + tan x) 6 2 2 = (7 tan x - 3 tan x) (1 + tan x) 6 2 2 = (7 tan x - 3 tan x) sec x \pi \pi 4 4 6 2 2 l1 = \int f (x)dx = \int (7 tan x - 3 tan x) sec xdx 0 0 \pi 7 3 7 tan x 3 tan x ∣4 = ( - )∣ = 1 - 1 = 0 7 3 ∣ 0 \pi \pi 4 4 6 2 2 I2 = \int xf (x)dx = \int x (7 tan x - 3 tan x) sec xdx 0 0 \pi \pi 4 7 3 7 3 x)∣ 4 = x (tan x - tan ∣ - \int 1 ⋅ (tan x - tan x) dx 0 0 \pi 4 3 2 2 = 0 - \int tan x (tan x - 1) (tan x + 1) dx 0 \pi \pi 4 6 x ∣ 4 4 tan x tan 3 5 2 = \int (tan x - tan x) sec xdx = - ∣ 0 4 6 ∣ 0 1 = 12 Hence 7I + 12I = 1 1 2
Correct Answer: 2