Integration techniques: JPJC 2024 Promo Q4

Consider $x=A(8x-8)+B$.

By comparing coefficient of $x$ and constant term: $A=\dfrac18$, $B=1$.

$$\begin{aligned} \int_0^1\frac{x}{4x^2-8x+5}\mathrm{d} x &=\frac18\int_0^1\frac{8x-8}{4x^2-8x+5}\mathrm{d} x +\int_0^1\frac1{4x^2-8x+5}\mathrm{d} x\\ &=\frac18\left[\ln(4x^2-8x+5)\right]_0^1 +\int_0^1\frac1{4[x^2-2x+(-1)^2-(-1)^2]+5}\mathrm{d} x\\ &=\frac18[\ln1-\ln5]+\int_0^1\frac1{4(x-1)^2+1}\mathrm{d} x\\ &=-\frac18\ln5+\int_0^1\frac1{4\left[(x-1)^2+\left(\frac12\right)^2\right]}\mathrm{d} x\\ &=-\frac18\ln5+\frac14\left[\frac1{\frac12}\tan^{-1}\left(\frac{x-1}{\frac12}\right)\right]_0^1\\ &=-\frac18\ln5+\frac12[\tan^{-1}(0)-\tan^{-1}(-2)]\\ &=-\frac18\ln5-\frac12\tan^{-1}(-2). \end{aligned}$$