Animated Solution for Mathematics - Definite Integration: The value of ${\int }_0^{\pi }\frac{e^{\cos x}\sin x}{(1+{\cos }^{2}x)(e^{\cos x}+e^{-\cos x})}dx$ is equal to

The integral provided is:

This expression appears intimidating due to the combination of exponential and trigonometric functions. However, in JEE Advanced mathematics, such complexity is often a signal to look for symmetry.

Whenever you encounter a definite integral with limits from $0$ to $a$, the 'King's Property' is your most powerful tool:

Applying this to our integral by replacing $x$ with $\pi -x$, we obtain:

Using the trigonometric identities $\sin (\pi -x)=\sin x$ and $\cos (\pi -x)=-\cos x$, the expression simplifies to:

Notice that the denominator remains unchanged. By adding the original integral $I$ to this new form, the exponential terms in the numerator and denominator cancel out:

This reduces the problem to a much simpler form:

Since the integrand is symmetric about $\pi /2$, we can simplify the integral to:

We now perform the substitution $t=\cos x$, which implies $dt=-\sin xdx$. Adjusting the limits, when $x=0$, $t=1$, and when $x=\pi /2$, $t=0$:

Evaluating this standard integral yields:

The final result is: