If ∫2cosx−sinx+λcosx+sinx−2dx=Aℓn|cosx+sinx&#x

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Question

If

\int \frac{2 c o s x - s i n x + λ}{c o s x + s i n x - 2} d x = A ℓ n | c o s x + s i n x - 2 | + B x + C

. Then the ordered triplet

A, B, λ

is

Options:

A . (12,32,−1)

B . (32,12,−1)

C . (12,−1,−32)

D . (32,−1,−12)

Answer: Option B
:
B

\frac{d}{d x} (A | c o s x + s i n x - 2 | + B x + C) = A \frac{c o s x - s i n x}{c o s x + s i n x - 2} + B = \frac{A c o s x + B c o s x - A s i n x + B s i n x - 2 B}{c o s x + s i n x - 2} ∴ A + B = 2, A + B = - 1, - 2 B = λ ∴ A = \frac{3}{2}, B = \frac{1}{2}, λ = - 1

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