Definite Integration
Trig Integral with Partial Fractions
nta_pyq_2023_jan
Grade 12
Question:
The value of $\displaystyle\int_{\pi/3}^{\pi/2}\dfrac{2+3\sin x}{\sin x(1+\cos x)}\,dx$ is equal to:
\dfrac{7}{2}-\sqrt{3}-\log_e\sqrt{3}
-2+3\sqrt{3}+\log_e\sqrt{3}
\dfrac{10}{3}-\sqrt{3}+\log_e\sqrt{3}
\dfrac{10}{3}-\sqrt{3}-\log_e\sqrt{3}
Step-by-Step Solution
Key Concept: Write $2+3\sin x=A\sin x(1+\cos x)+B\sin x+C(1+\cos x)$ or decompose differently. Split: $\frac{2+3\sin x}{\sin x(1+\cos x)}=\frac{2}{\sin x(1+\cos x)}+\frac{3}{1+\cos x}$.
$\dfrac{10}{3}-\sqrt{3}-\log_e\sqrt{3}$.
Correct Answer: 4