Express $1+4x$ in the form $A(2x-2)+B$, where $A$ and $B$ are constants to be determine

Question

Express $1+4x$ in the form $A(2x-2)+B$, where $A$ and $B$ are constants to be determined.[1] Hence, find $\displaystyle\int \frac{1+4x}{x^2-2x+5} \, \mathrm{d}x$[4] Find the exact value of $\displaystyle\int_{0}^{\frac{\pi}{3}} x \sin 2x \, \mathrm{d}x$[3]

Accepted Answer

(a)(i) $A=2,\ B=5$. (a)(ii) $2\ln(x^2-2x+5)+\frac52\tan^{-1}\!\left(\frac{x-1}{2}\right)+C$. (b) $\frac\pi{12}+\frac{\sqrt3}{8}$.

2025 VJC P1 Q10

2025 VJC P1 Q10

Solution:

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