Animated Solution for Mathematics - Definite Integration: If ${\int }_0^{\pi /2}\frac{\cot x}{\cot x+\csc x}dx=m(\pi +n)$, then $m\cdot n$ is equal to :

The integral is given by:

This expression appears complex, but we can simplify it by expressing the trigonometric functions in terms of $\sin x$ and $\cos x$. We know that $\cot x=\frac{\cos x}{\sin x}$ and $\csc x=\frac{1}{\sin x}$.

Substituting these into the integrand, we get:

By multiplying both the numerator and the denominator by $\sin x$, the terms simplify significantly. We are left with the much more manageable integrand:

Now we must integrate $\frac{\cos x}{1+\cos x}$. To handle the $+1$ in the denominator, we utilize standard trigonometric half-angle identities.

Recall that $1+\cos x=2{\cos }^2(x/2)$ and $\cos x=2{\cos }^2(x/2)-1$. Substituting these into our integral yields:

By splitting the fraction, we obtain:

We can now integrate the expression term by term. The integral of $1$ is $x$, and the integral of ${\sec }^2(x/2)$ is $2\tan (x/2)$.

Applying the coefficient $\frac{1}{2}$, the anti-derivative becomes $x-\tan (x/2)$. Evaluating this from $0$ to $\pi /2$:

Since $\tan (\pi /4)=1$ and $\tan (0)=0$, the result is:

The problem requires the result in the form $m(\pi +n)$. We factor out $\frac{1}{2}$ from our result:

By comparing this to $m(\pi +n)$, we identify $m=\frac{1}{2}$ and $n=-2$. Therefore, the final product is: