Animated Solution for Mathematics - Definite Integration: Evaluate ${\int }_0^{\pi }{e}^{|\cos x|}\left(2\sin \left(\frac{1}{2}\cos x\right)+3\cos \left(\frac{1}{2}\cos x\right)\right)\sin xdx$.

Welcome, fellow traveler of the mathematical landscape. Today, we stand before an integral that, at first glance, seems designed to intimidate:

It contains exponentials, absolute values, and nested trigonometric functions. But remember, in the world of JEE Advanced, complexity is often just a mask for elegance. Our job is to peel back that mask.

Whenever you see a function and its derivative sitting together, your intuition should immediately fire. Look at the term $\sin xdx$ at the end of our integral; it is the derivative of $\cos x$, which appears repeatedly inside the other functions.

Let $t=\cos x$. Then, $dt=-\sin xdx$, or $\sin xdx=-dt$. This substitution is the key that unlocks the door.

When $x=0$, $t=\cos (0)=1$. When $x=\pi$, $t=\cos (\pi )=-1$. Our integral transforms into:

Now, let us clean up. That negative sign from $-dt$ is a gift; we can use it to flip the limits of integration from $[1,-1]$ to $[-1,1]$.

Now we have:

Whenever you see limits from $-a$ to $a$, your brain should scream: "Check for even and odd functions!" This is a favorite JEE tactic.

Let us split the integral into two parts:

For $I_1$, the function $f_1(t)=2e^{|t|}\sin (\frac{t}{2})$ is odd because $f_1(-t)=-f_1(t)$. The integral of an odd function over symmetric limits is zero, as the positive and negative areas cancel out perfectly. We are left only with $I_2$.

Since $f_2(t)=3e^{|t|}\cos (\frac{t}{2})$ is an even function, we can simplify $I_2$ to:

We are now left with a standard integral. We use the standard formula:

Here, $a=1$ and $b=1/2$. The denominator $a^2+b^2=1+1/4=5/4$. The integral becomes:

Evaluating at the limits, we get:

This simplifies to our final, elegant answer:

See? The complexity was just a test of your patience and your ability to spot the underlying symmetry. Keep practicing, and these patterns will become second nature!