Q90 / 150 · MHT CET 2019 (Shift 1) · Mathematics

$\int \frac{d x}{(s i n x + c o s x) (2 c o s x + s i n x)} =$

Solution

Let

l = \int \frac{d x}{(s i n x + c o s x) (2 c o s x + s i n x)}

= \int \frac{{s e c}^{2} x}{(t a n x + 1) (2 + t a n x)} d x

Put

t a n x = t

\Rightarrow {s e c}^{2} x d x = d t \int \frac{d t}{(t + 1) (t + 2)}

H e r e

\frac{1}{(t + 1) (t + 2)} = \frac{A}{t + 1} + \frac{B}{t + 2}

\Rightarrow 1 = A (t + 2) + B (t + 1)

Put

t = - 2

∴ 1 = B (- 1) \Rightarrow B = - 1

∴ P u t t + 1 = 0 \Rightarrow t = - 1

∴ 1 = A (1) \Rightarrow A = 1

N o w

\int \frac{d t}{(t + 1) (t + 2)} = \int \frac{1}{t + 1} d t = - \int \frac{1}{t + 2} d t

= \log (t + 1) - \log (t + 2) + C

= \log (\frac{t + 1}{t + 2}) + C

= \log (\frac{t a n x + 1}{t a n x + 2}) + C

Hence, the correct answer is option: 'D'.