Step-by-step explanation:

\int_0^{\pi} \frac{x \sec(x) : dx }{ \tan(x) + \sec(x) }

we can use the substitution u = tan(x) + sec(x). This substitution will allow us to rewrite the integral in terms of u, which will make it easier to solve.

When we make this substitution, we have to remember to use the chain rule to express the derivative of x in terms of u. The chain rule tells us that if y = f(u) and u = g(x), then dy/dx = dy/du * du/dx. In this case, y = x and u = tan(x) + sec(x), so we can write:

dy/dx = dx/du * du/dx

= 1 * du/dx

We also know that du/dx = sec^2(x), so we can substitute this value into the equation above to get:

dy/dx = sec^2(x)

Now, we can substitute u = tan(x) + sec(x) into the original integral and use the chain rule to express the derivative of x in terms of u:

\int_0^{\pi} \frac{x \sec(x) : dx }{ \tan(x) + \sec(x) } = \int_0^{\pi} \frac{u : du}{u}

The integral on the right side is now easy to solve:

\int_0^{\pi} \frac{u : du}{u} = \int_0^{\pi} du = [u]_0^{\pi} = [\pi] - [0] = \pi

Therefore, the value of the original integral is

\int_0^{\pi} \frac{x \sec(x) : dx }{ \tan(x) + \sec(x) } = \pi