Question

\( \int \frac{(5 \sin \theta - 2) \cos \theta}{(5 - \cos^2 \theta - 4 \sin \theta)} d\theta = \)}

• A. \( (\log 5 \sin \theta - 2) + c \)
• B. \( 5 \log(5 \sin \theta - 2) - \frac{8}{(\sin \theta - 2)} + c \)
• C. \( 5 \log |\sin \theta - 2| + \frac{8}{2 - \sin \theta} + c \)
• D. \( \log (5 \sin \theta - 2) + \frac{1}{(\sin \theta - 2)} + c \)

Hint:

When you see $\cos \theta$ in the numerator and only $\sin \theta$ or $\cos^2 \theta$ in the denominator, always substitute $u = \sin \theta$.

Answer 1

The Correct Option is C

Step 1: Substitution
Let $u = \sin \theta \implies du = \cos \theta d\theta$.
The denominator $5 - (1 - \sin^2 \theta) - 4 \sin \theta = u^2 - 4u + 4 = (u-2)^2$.
Step 2: Simplify Integral
$\int \frac{5u - 2}{(u-2)^2} du = \int \frac{5(u-2) + 8}{(u-2)^2} du$.
$= \int (\frac{5}{u-2} + \frac{8}{(u-2)^2}) du$.
Step 3: Integration
$5 \log |u-2| - \frac{8}{u-2} + c$.
Step 4: Back Substitution
$5 \log |\sin \theta - 2| + \frac{8}{2 - \sin \theta} + c$.
Final Answer:(C)