200 days streak special 5

Let $J = \displaystyle \int_0^2 \dfrac{{\text{d}x}}{(17 + 8x - 4x^2)(e^{6(1-x)} + 1)}$ be the required integral.

Using the property $\displaystyle \int_a^b f(t) \text{d}t =\displaystyle \int_a^b f(a + b - t) \text{d}t$ , we get

$2J = \int_0^2 \dfrac{{\text{d}x}}{(17 + 8x - 4x^2)(e^{6(1-x)} + 1)} + \int_0^2 \dfrac{{\text{d}x}}{(17 + 8x - 4x^2)(e^{-6(1-x)} + 1)}$ $2J = \int_0^2 \dfrac{e^{6(1-x)} + 1 }{(17 + 8x - 4x^2)(e^{6(1-x)} + 1)} \text{d}x = \int_0^2 \dfrac{\text{d}x}{17 + 8x - 4x^2}$

This integral can be written as $\displaystyle \int_0^2 \dfrac{\text{d}x}{21 - 4(x-1)^2}$ which can be evaluated using standard techniques to give $J = \boxed{\dfrac{1}{4\sqrt{21}} \ln \left(\dfrac{\sqrt{21} + 2}{\sqrt{21} - 2}\right)}$