Indefinite Integration
Integration by Decomposition into Numerator Form
DAILY_CHALLENGE
Grade 12
Question:
Let $\displaystyle\int\frac{2-\tan x}{3+\tan x}\,dx=\frac{1}{2}\!\left(\alpha x+\log_e|\beta\sin x+\gamma\cos x|\right)+C$, where $C$ is the constant of integration. Then $\alpha+\dfrac{\gamma}{\beta}$ is equal to:
Step-by-Step Solution
Key Concept: Write $\frac{2\cos x-\sin x}{3\cos x+\sin x}=A+B\cdot\frac{d}{dx}(3\cos x+\sin x)/(3\cos x+\sin x)$. Decompose numerator: $2\cos x-\sin x=A(3\cos x+\sin x)+B(\cos x-3\sin x)$.
$\alpha=1$, $\beta=1$, $\gamma=3$. $\alpha+\gamma/\beta=4$.
Correct Answer: 2