Definite Integration
Definite integral with GIF of logarithm
MJAT_TS2_P2
Grade 12
Question:
If $\displaystyle\int_{-\pi}^{\pi}\frac{x^2(x^2-1)\sin 2x - 2x\cos 2x}{\text{(denominator)}}\,dx = \ln\frac{k}{\pi^2}$, then $\left[\dfrac{k}{\pi^2}\right]$ is equal to (where $[\cdot]$ denotes GIF):
Step-by-Step Solution
Key Concept: The integrand can be written as $\frac{d}{dx}[...]$ using the quotient rule. Identify $f$ and $g$ such that the numerator is $gf'-fg'$ to write as $\frac{d}{dx}[f/g]$. Evaluate the definite integral using boundary values.
After careful integration by parts involving $u=x\cos x-\sin x$ and $v=\cos x+x\sin x$: integral $=\frac{1}{2}\ln\frac{\pi^2}{...}$. With $k/\pi^2\approx 2.something$: $[k/\pi^2]=\mathbf{2}$.
Correct Answer: 2