Integral Calculus-1
Integral Calculus-1
Allen Star Batch
Grade 12
Question:
Let $f$ is a differentiable function such that $f(x) = x^2 + \int_0^x e^{-t}f(x-t)dt$, then:
$\int e^x f(x)dx = \frac{e^x x^2}{3} + c$
$f(x)$ is neither even nor odd
$\int e^x f(x)dx = \frac{e^x x^2}{2} + c$
$f(3) = 18$
Step-by-Step Solution
Key Concept: Differentiate the integral equation using Leibniz rule to convert it into a differential equation, then solve using integration and initial conditions.
Given $f(x) = x^2 + \int_0^x e^{-t}f(x-t)dt$, differentiate both sides to obtain $f'(x) = 2x + e^{-x}[e^xf(x)] - \int_0^x e^{-t}f(t)dt$. This simplifies to $f(x) + f'(x) = x^2 + 2x$. Integrating yields $f(x) = \frac{x^3}{3} + x^2 + C$. Using the initial condition $f(0) = 0$, we get $C = 0$, so $f(x) = \frac{x^3}{3} + x^2$.
Correct Answer: 1,2,4