Integral Calculus
Definite integral of odd function via determinant
MMTS_Full_Test_06
Grade 12
Question:
If $f(x) = \begin{vmatrix}\cos x & e^{x^2} & 2x\cos^2(x/2)\\ x^2 & \sec x & \sin x+x^3\\ 1 & 2 & x+\tan x\end{vmatrix}$ and $\displaystyle\int_{-\pi/2}^{\pi/2}(1+x^4)(f(x)+f''(x))\,dx = 2\lambda+3$, then $\lambda$ is
(A) $\dfrac{1}{2}$
(B) $\dfrac{3}{2}$
(C) $-\dfrac{1}{2}$
(D) $-\dfrac{3}{2}$
Step-by-Step Solution
Key Concept: Check the parity of $f(x)$: examine each row/column when $x\to-x$. If $f(x)$ is odd, then $f(x)+f''(x)$ is also odd.
$f$ odd $\Rightarrow f+f''$ odd $\Rightarrow$ integral $=0=2\lambda+3 \Rightarrow \lambda=-3/2$.
Correct Answer: (D) $-\dfrac{3}{2}$