Animated Solution for Mathematics - Differential Equations: Let $f:[0,2]\to R$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0)=1$. Let $F(x)={\int }_0^{x^2}f(\sqrt{t})dt$ for $x\in [0,2]$. If $F^{\prime }(x)=f^{\prime }(x)$ for all $x\in (0,2)$, then $f(2)$ equals

Imagine you are standing before a complex integral, $F(x)={\int }_0^{x^2}f(\sqrt{t})dt$. It looks intimidating, but in the world of JEE Advanced, intimidation is just a mask for elegance.

We are given that $F^{\prime }(x)=f^{\prime }(x)$, a powerful clue that links the integral function directly to the derivative of the function inside it. To unlock this, we utilize the Leibniz Rule.

This rule is your secret weapon for differentiating integrals where the limits are functions of $x$. It states that the derivative of an integral is the integrand evaluated at the upper limit multiplied by the derivative of that limit, minus the integrand evaluated at the lower limit multiplied by the derivative of that lower limit.

Applying this to our function, we get:

Since $\sqrt{x^2}=x$ for $x\in [0,2]$, this simplifies beautifully to:

Now, the problem becomes a dance of algebra. Given $F^{\prime }(x)=f^{\prime }(x)$, we can write:

This is a classic first-order separable differential equation. By rearranging the terms, we obtain:

This is the moment where calculus feels like magic. We integrate both sides with respect to $x$:

The left side becomes $\ln |f(x)|$, and the right side becomes $x^2+C$. Exponentiating both sides, we arrive at $f(x)=e^{x^2+C}$, which we can write as:

where $A=e^C$.

We are given the initial condition $f(0)=1$. Plugging this into our equation, $1=A{e}^{0^2}$, which forces $A=1$.

So, our function is simply $f(x)=e^{x^2}$. Since $F^{\prime }(x)=f^{\prime }(x)$, we know $F(x)=f(x)+K$.

We know $F(0)={\int }_0^{0}f(\sqrt{t})dt=0$. Using $f(0)=1$, we find $0=1+K$, so $K=-1$.

Thus, the general form of our function is:

Finally, evaluating at $x=2$, we get:

There it is—the elegance of the solution. You have navigated the integral, solved the differential equation, and found the constant. That is the essence of JEE Advanced mathematics.