JEE Main · 2012 · 29 Apr 2012 · Offline · Indefinite Integrals

If the integral

\int \frac{5 t a n x}{t a n x - 2} d x = x + a ln ∣ sin x - 2 cos x ∣ + k

, then a is equal to

Answer: (D)

Solution

Write the integrand as

\frac{5 sin x}{sin x - 2 cos x}

and express the numerator as a combination of the denominator and its derivative:

5 sin x = A (sin x - 2 cos x) + B (cos x + 2 sin x)

. Matching coefficients gives

A + 2 B = 5

and

- 2 A + B = 0

, so

A = 1

and

B = 2

. The integral then splits as

\int d x + 2 \int \frac{cos x + 2 sin x}{sin x - 2 cos x} d x = x + 2 ln ∣ sin x - 2 cos x ∣ + k

, so

a = 2

JEE Main · 2012 · 29 Apr 2012 · Offline · Single correct option

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