# Integral $\int_0^{\pi/2} x\cot(x)dx$, Differntiation wrt parameter only.

Integrate using differentiation wrt parameter only.

$$\int_0^{\pi/2} x\cot(x)dx$$

We can express this as

$$\int_0^{\pi/2} x\cdot\frac{\cos(x)}{\sin(x)}dx$$

Notice we can write $u=\sin(x)$ to start but I am not sure if that will do us any good.

If we use $\xi$ as a parameter, the answer is of the form.$$\lim_{\xi \to 1}I(\xi)=\lim_{\xi \to 1}\frac{\pi}{2}\ln(\xi+1)$$

NOTE:Only use differentiation with respect to parameter.

#### Solutions Collecting From Web of "Integral $\int_0^{\pi/2} x\cot(x)dx$, Differntiation wrt parameter only."

found the way to go…We can want to solve

$$I=\int_0^{\pi/2}x\cot(x) dx,$$
so we introduce a parameter $\xi$ by writing

$$I(\xi)=\int_0^{\pi/2} \frac{\tan^{-1}(\xi\tan(x))}{\tan(x)} dx$$
and in the limit $\xi \to 1$ we recover I. Taking a derivative we obtain
$$I'(\xi)=\frac{d}{d\xi}\int_0^{\pi/2} \frac{\tan^{-1}(\xi\tan(x))}{\tan(x)} dx=\int_0^{\pi/2}\frac{\partial}{\partial \xi} \bigg(\frac{\tan^{-1}(\xi\tan(x))}{\tan(x)} \bigg)dx$$
Now we take the derivative to obtain
$$I'(\xi)=\int_0^{\pi/2} \frac{dx}{\big(\xi\tan(x)\big)^2+1} =\frac{\pi}{2(\xi+1)}.$$
We now integrate our result wrt $\xi$ and realizing the constant of integration is zero, we obtain
$$I(\xi)=\frac{\pi}{2}\ln(\xi+1).$$
Taking the limit as $\xi \to 1$ we obtain
$$\lim_{\xi \to 1} I(\xi)=\lim_{\xi \to 1} \frac{\pi}{2}\ln(\xi+1)=\frac{\pi \ln(2)}{2}.$$
Thus we have shown that
$${\boxed{I=\frac{\pi\ln(2)}{2}}}$$