Definite Integration
Integral of log of trigonometric function over period
MJAT_TS3_P2
Grade 12
Question:
If $\displaystyle\int_0^{4\pi}\ln\left|\frac{1}{3}\sin x+\sqrt{3}\cos x\right|\,dx = k\pi\ln 7$, then the value of $k$ is:
Step-by-Step Solution
Key Concept: Write $\frac{1}{3}\sin x+\sqrt{3}\cos x = \frac{2\sqrt{3}}{3}\sin(x+\alpha)$ (or similar R-form). Then $\int_0^{4\pi}\ln|R\sin(x+\alpha)|dx=4\int_0^\pi(\ln R+\ln|\sin x|)dx=4(\pi\ln R+(-\pi\ln 2))$. Choose $R$ and $\alpha$ appropriately.
$k=\mathbf{4}$.
Correct Answer: B