Animated Solution for Mathematics - Definite Integration: If ${\int }_0^{\pi /3}\frac{\tan \theta }{\sqrt{2k\sec \theta }}d\theta =1-\frac{1}{\sqrt{2}}(k>0)$, then the value of $k$ is :

Welcome, future engineer. Today we tackle an integral that looks like a monster but is actually a masterpiece of symmetry.

When you first see the expression

it is natural to feel a moment of hesitation. But remember, in the world of JEE Advanced, complexity is often just a veil for elegance.

The first step is to strip away the complexity. We see $\tan \theta$ and $\sec \theta$ dancing together.

The golden rule here is to bring everything back to the fundamental language of trigonometry: $\sin \theta$ and $\cos \theta$. We know that $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $\sec \theta =\frac{1}{\cos \theta }$.

Substituting these into our integrand, we get:

This simplifies beautifully. The $\cos \theta$ terms interact, and we are left with:

Suddenly, the monster is tamed.

Now, look at the structure. We have $\sin \theta$ in the numerator and $\sqrt{\cos \theta }$ in the denominator. This is a classic setup for the substitution method.

Let $u=\cos \theta$. Then, the differential $du=-\sin \theta d\theta$. This means $\sin \theta d\theta =-du$.

This is the 'Aha!' moment where the path forward becomes clear.

A common trap for students is forgetting to update the limits of integration. When $\theta =0$, $u=\cos (0)=1$. When $\theta =\pi /3$, $u=\cos (\pi /3)=1/2$.

Our integral now becomes:

By using the negative sign to flip the limits, we get:

Now, we apply the power rule: $\int u^{n}du=\frac{u^{n+1}}{n+1}$. For $n=-1/2$, this gives $2\sqrt{u}$.

Evaluating this from $1/2$ to $1$, we get $2(1-\sqrt{1/2})=2-\sqrt{2}$. Combining this with our constant, we have:

Equating this to the original right-hand side, $1-\frac{1}{\sqrt{2}}=\frac{\sqrt{2}-1}{\sqrt{2}}$, we see that:

Therefore, $k=2$. It is a perfect, clean result. Keep practicing, and you will find that even the most intimidating problems have a logical, beautiful heart.