Animated Solution for Mathematics - Definite Integration: The value of the integral ${\int }_{1/2}^2\frac{{\tan }^{-1}x}{x}dx$ is equal to

Unlocking the Integral

Welcome, future engineer. Today, we are going to dismantle a problem that, at first glance, looks like a nightmare of transcendental functions. We are tasked with evaluating the integral $I={\int }_{1/2}^2\frac{{\tan }^{-1}x}{x}dx$.

If you try to find an antiderivative for $\frac{{\tan }^{-1}x}{x}$ using standard methods, you will quickly find yourself in a labyrinth with no exit. But in the JEE Advanced arena, when a problem seems impossible to solve directly, it is usually a sign that there is a hidden symmetry waiting to be exploited.

The Reciprocal Insight

Look closely at the limits of integration: $1/2$ and $2$. They are reciprocals. This is not a coincidence; it is a deliberate invitation.

Whenever you encounter limits that are reciprocals, the substitution $x=\frac{1}{t}$ is your golden key. Let us apply this transformation. If $x=\frac{1}{t}$, then $dx=-\frac{1}{t^2}dt$.

Now, let us transform the limits: when $x=1/2$, $t=2$, and when $x=2$, $t=1/2$. Our integral $I$ transforms into:

Notice the negative sign from the differential $dx$. We can use that negative sign to flip the limits back to their original order, from $1/2$ to $2$. After simplifying the algebra, we get:

The Identity Magic

Here is where the elegance of mathematics shines. We know that for $t>0$, the inverse trigonometric identity ${\tan }^{-1}(1/t)={\cot }^{-1}(t)$ holds true. Our integral now looks like this:

Since $t$ is just a dummy variable, we can rename it back to $x$ without changing the value of the integral. We now have two different expressions for the same value $I$: the original one and this new one involving ${\cot }^{-1}x$.

This is the moment of truth. Let us add them together:

The Grand Simplification

Because the limits and the denominators are identical, we can merge these integrals into one:

Recall the fundamental identity: ${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}$. The entire numerator collapses into a simple constant!

Our terrifying integral has suddenly become trivial:

The Final Stretch

The integral of $1/x$ is simply $\ln |x|$. Evaluating this from $1/2$ to $2$, we get:

Since $\ln (1/2)=-\ln 2$, the expression becomes $\frac{\pi }{2}(\ln 2-(-\ln 2))=\frac{\pi }{2}(2\ln 2)$. Therefore, $2I=\pi \ln 2$, which leads us to our final answer:

This problem is a masterclass in why we study properties of definite integrals. We didn't need to find a complex antiderivative; we simply needed to see the symmetry, apply the right substitution, and let the identities do the heavy lifting. Keep this strategy in your arsenal—it is a weapon that will serve you well in the exam hall.