Animated Solution for Mathematics - Indefinite Integration: $\int \frac{\sin (5x/2)}{\sin (x/2)}dx$ is equal to : (where c is a constant of integration)

Welcome, fellow traveler on the path to JEE excellence. Today, we are going to tackle a problem that looks intimidating at first glance: the integral of a ratio of sine functions, $I=\int \frac{\sin (5x/2)}{\sin (x/2)}dx$.

When you see a fraction like this, your first instinct might be to panic. But I want you to take a deep breath. In mathematics, as in life, when a problem seems too complex to handle as a whole, we break it down into smaller, manageable pieces.

We are going to perform a 'surgical' deconstruction of this integrand.

The Strategic Split

Look closely at the angles. We have $5x/2$ in the numerator and $x/2$ in the denominator. The secret to unlocking this problem lies in the relationship between these two values.

If we express the larger angle in terms of the smaller one, we can create a bridge. Let us write $5x/2$ as $2x+x/2$.

Because the compound angle identity $\sin (A+B)=\sin A\cos B+\cos A\sin B$ is waiting to help us, we set $A=2x$ and $B=x/2$. We can expand our numerator into:

The Great Cancellation

Now, watch the magic happen. When we place this expansion back over our denominator, $\sin (x/2)$, the expression splits into two distinct parts:

Notice how the second term simplifies instantly to $\cos 2x$. That is a gift!

The first term, however, is $\sin 2x\cot (x/2)$. We know that $\sin 2x=2\sin x\cos x$, and further, $\sin x=2\sin (x/2)\cos (x/2)$. Substituting these into our first term gives us:

The Final Simplification

We are almost there. We have $4{\cos }^2(x/2)\cos x+\cos 2x$. That ${\cos }^2(x/2)$ is still a bit messy.

Let us use the power-reduction identity $2{\cos }^2\theta =1+\cos 2\theta$. With $\theta =x/2$, this becomes $2{\cos }^2(x/2)=1+\cos x$.

Substituting this back, our expression transforms into $2(1+\cos x)\cos x+\cos 2x$, which expands to $2\cos x+2{\cos }^{2}x+\cos 2x$.

One last step! We have another squared term, $2{\cos }^{2}x$. Using the same power-reduction identity, $2{\cos }^{2}x=1+\cos 2x$. Our final integrand is now a beautiful, linear sequence:

Now, integration becomes a joy. We simply integrate term by term:

Simplifying this, we arrive at our final answer:

See? The complexity was just a mask. By systematically applying identities, we stripped away the layers until only the simple, elegant truth remained. Keep this mindset for your JEE preparation: never fear the complexity; just look for the identity that will set you free.