Definite Integration
Definite Integral — tan Substitution
nta_pyq_2024_apr
Grade 12
Question:
If $\int_0^{\pi/2}\dfrac{\sin^2 x}{1+\sin x\cos x}\,dx=\dfrac{1}{a}\log_e\left(\dfrac{a}{3}\right)+\dfrac{\pi}{b\sqrt{3}}$, where $a,b\in\mathbb{N}$, then $a+b$ is equal to ________.
Step-by-Step Solution
Key Concept: Write $\sin^2x=\frac{1-\cos2x}{2}$ and $1+\sin x\cos x=1+\frac{\sin2x}{2}$. Split into $I_1-I_2$. Use $t=\tan x$ substitution for $I_1$.
$a=2$, $b=6$. $a+b=8$.
Correct Answer: 8