Integral Calculus
Definite integral with GIF; GIF of sum
MJMT_Full_Test_11
Grade 12
Question:
For positive integer $n$, let $I_n=\displaystyle\int_{-\pi}^{\pi}\!\left(\frac{\pi}{2}-|x|\right)\cos nx\,dx$. Find $[I_1+I_2+I_3+I_4]$ (GIF).
Step-by-Step Solution
Key Concept: By symmetry (even integrand): $I_n=2\int_0^\pi(\pi/2-x)\cos nx\,dx$. IBP: $I_n=2(1-\cos n\pi)/n^2$. Odd $n$: $I_n=4/n^2$; even $n$: $I_n=0$. Sum $=I_1+I_3=4+4/9=40/9\approx4.44$.
$[40/9]=4$.
Correct Answer: 4