JEE PYQ: Indefinite Integration Question 12
Question 12 - 2020 (05 Sep Shift 2)
If $\int \frac{\cos\theta}{5 + 7\sin\theta - 2\cos^2\theta},d\theta = A\log_e|B(\theta)| + C$, where $C$ is a constant of integration, then $\frac{B(\theta)}{A}$ can be:
(1) $\frac{2\sin\theta + 1}{\sin\theta + 3}$
(2) $\frac{2\sin\theta + 1}{5(\sin\theta + 3)}$
(3) $\frac{5(\sin\theta + 3)}{2\sin\theta + 1}$
(4) $\frac{5(2\sin\theta + 1)}{\sin\theta + 3}$
Show Answer
Answer: (4)
Solution
Let $\sin\theta = t$, $\cos\theta,d\theta = dt$. Denominator: $5 + 7t - 2(1-t^2) = 2t^2 + 7t + 3 = (2t+1)(t+3)$. Partial fractions: $\frac{1}{5}\ln\left|\frac{2\sin\theta+1}{\sin\theta+3}\right| + C$. So $A = \frac{1}{5}$, $B(\theta) = \frac{2\sin\theta+1}{\sin\theta+3}$, and $\frac{B(\theta)}{A} = \frac{5(2\sin\theta+1)}{\sin\theta+3}$.
BETA
