Animated Solution for Mathematics - Definite Integration: ${\lim }_{x\to 0}\frac{{\int }_0^{x}t\sin (10t)dt}{x}$ is equal to

When you look at this expression, your eyes might be drawn to the integral in the numerator. It is natural to feel a surge of anxiety—do we need to perform integration by parts? Do we need to find a complex antiderivative?

Take a deep breath. In mathematics, as in life, the most complex-looking problems often have the most elegant solutions if we just pause to observe the structure.

Let us define our numerator as a function of $x$, say $F(x)={\int }_0^{x}t\sin (10t)dt$. As $x$ approaches $0$, the upper limit of our integral collapses onto the lower limit. The area under the curve from $0$ to $0$ is, by definition, $0$.

Simultaneously, our denominator $x$ is also approaching $0$. We have arrived at the classic $\frac{0}{0}$ indeterminate form. This is not a dead end; it is an invitation to use one of the most powerful tools in our arsenal: L'Hospital's Rule.

L'Hospital's Rule tells us that if we have a $\frac{0}{0}$ form, the limit of the ratio of two functions is equal to the limit of the ratio of their derivatives. But how do we differentiate an integral?

This is where the Newton-Leibniz Formula (or the Fundamental Theorem of Calculus) comes to our rescue. It states that the derivative of an integral with respect to its upper limit is simply the integrand evaluated at that limit:

In our case, $f(t)=t\sin (10t)$. Therefore, the derivative of our numerator is simply $x\sin (10x)$. It is as if the integral was just a shell, and the derivative stripped it away to reveal the core function underneath.

Now, let us look at our denominator. The derivative of $x$ with respect to $x$ is simply $1$. Our limit has transformed from a terrifying integral into a simple algebraic expression:

As $x$ approaches $0$, we substitute the value directly. We get $0\cdot \sin (10\cdot 0)$, which is $0\cdot \sin (0)$.

Since $\sin (0)=0$, we are left with $0\cdot 0$, which equals $0$.

I want you to reflect on what just happened. We started with an integral that seemed to demand heavy lifting, but by understanding the geometric nature of the limit and the relationship between integration and differentiation, we reduced it to a trivial calculation.

This is the beauty of the JEE Advanced syllabus—it is not about memorizing formulas; it is about recognizing the underlying rhythm of the math. You have the tools, you have the logic, and now you have the experience. Keep pushing, keep questioning, and keep finding the elegance in the chaos.