Integral Calculus
Integral equation; ODE via differentiation
MMTS_Full_Test_12
Grade 12
Question:
If $x\displaystyle\int_0^x f(t)\,dt = (x+1)\int_0^x tf(t)\,dt$ for $x>0$, and $f(1)=\dfrac{1}{e}$, then $f(-1)$ is
(A) $-\dfrac{1}{e^{1/2}}$
(B) $-\dfrac{1}{e}$
(C) $e$
(D) $-e$
Step-by-Step Solution
Key Concept: Differentiate both sides with respect to $x$. Rearrange to get a separable/linear ODE for $f(x)$.
$f(x)=x^{-3}e^{-1/x}$. $f(-1)=(-1)^{-3}e^{1}=-e$.
Correct Answer: (D) $-e$