Express $\frac{13x - 2}{(3 - x)(1 + 4x^2)}$ in partial fractions.[3] Hence, find the exact value o

Question

Express $\frac{13x - 2}{(3 - x)(1 + 4x^2)}$ in partial fractions.[3] Hence, find the exact value of $\int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{13x - 2}{(3 - x)(1 + 4x^2)} \,\mathrm{d}x$, leaving your answer in the form $\ln p + q\pi$, where $p$ and $q$ are rational numbers to be determined.[4] You are given that $I = \int \frac{a^2 - x^2}{(a^2 + x^2)^2} \,\mathrm{d}x$, where $a$ is a positive constant. Use the substitution $x = a \tan t$, where $-\frac{\pi}{2} < t < \frac{\pi}{2}$, to find an expression for $I$ in terms of $a$, leaving your answer in a non-trigonometric form.[5]

Accepted Answer

(a)(i) $\dfrac1{3-x}+\dfrac{4x-1}{1+4x^2}$. (ii) $\ln\dfrac75-\dfrac\pi4$, with $p=\dfrac75,q=-\dfrac14$. (b) $I=\dfrac{x}{a^2+x^2}+C$.

2024 NJC Promo Q11

2024 NJC Promo Q11

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