Animated Solution for Mathematics - Definite Integration: Let $f:(0,\infty )\to R$ and $F(x)={\int }_0^{x}f(t)dt$. If $F(x^2)=x^2(1+x)$, then $f(4)$ equals

Imagine you are standing on the edge of a vast, unknown landscape. You are given a function $F(x)={\int }_0^{x}f(t)dt$, which represents the area under the curve $y=f(t)$ from $0$ to $x$.

In our problem, we are looking at $F(x^2)={\int }_0^{x^2}f(t)dt$. This is the area under the curve up to a variable limit $x^2$. It is a dynamic, shifting boundary, and our goal is to uncover the hidden function $f(t)$ that defines this area.

Our Logic Bridge

How do we connect the area function $F$ to the function $f$? This is where the Fundamental Theorem of Calculus (FTC) becomes our most powerful tool.

It tells us that the derivative of an area function with respect to its upper limit is simply the value of the function at that limit. Mathematically, this is expressed as:

This is the bridge we need. It allows us to transform an integral equation into a differential one, which is much easier to handle.

We are given the relation $F(x^2)=x^2(1+x)$. Before we dive into the calculus, let us simplify the right-hand side.

By distributing $x^2$, we get:

This simple algebraic step is crucial. It transforms a product into a sum, making the subsequent differentiation much cleaner and less prone to errors.

The Core Difficulty

Now, we must differentiate both sides with respect to $x$. We are differentiating $F(x^2)$ with respect to $x$.

Since the inner function is $x^2$, we must use the Chain Rule:

The derivative of $x^2$ is $2x$. On the right side, the derivative of $x^2+x^3$ is $2x+3x^2$.

Putting it all together, we get:

Now, we use our FTC relation: $F^{\prime }(x^2)=f(x^2)$. Substituting this into our equation, we get:

To isolate $f(x^2)$, we divide both sides by $2x$. Since the domain is $(0,\infty )$, $x$ is never zero, so this division is perfectly valid.

We obtain:

We are asked to find $f(4)$. To get this, we set the argument $x^2=4$. Since $x$ must be positive, we take $x=2$.

Substituting $x=2$ into our expression, we get:

The final answer is 4. This problem is a perfect example of how the Fundamental Theorem of Calculus and the Chain Rule work in harmony to solve complex problems.