Definite Integration
Partial Fractions in Definite Integral
nta_pyq_2024_apr
Grade 12
Question:
The integral $\int_0^{\pi/4}\dfrac{136\sin x}{3\sin x+5\cos x}\,dx$ is equal to:
$3\pi-50\log_e2+20\log_e5$
$3\pi-25\log_e2+10\log_e5$
$3\pi-10\log_e(2\sqrt{2})+10\log_e5$
$3\pi-30\log_e2+20\log_e5$
Step-by-Step Solution
Key Concept: Write $136\sin x=A(3\sin x+5\cos x)+B(3\cos x-5\sin x)$. Solve: $A=12$, $B=-20$. Integrate each part.
$A=12$, $B=-20$. $I=3\pi-50\log_e2+20\log_e5$.
Correct Answer: 1