Evaluate $\displaystyle\int_{-a}^{a}{\left| {{x}^{2}}+1-{{\mathrm{e}}^{x}} \right|\,}\mathrm{d}x$ wh

Question

Evaluate $\displaystyle\int_{-a}^{a}{\left| {{x}^{2}}+1-{{\mathrm{e}}^{x}} ight|\,}\mathrm{d}x$ where $a>0$, giving your answer in terms of $a$. [3] By means of substituting $x=	an 	heta $, or otherwise, show that $\displaystyle\int{\frac{{{x}^{2}}}{{{\left( 1+{{x}^{2}} ight)}^{2}}}\,\mathrm{d}x}=\frac{1}{2}\left( {{	an }^{-1}}x+\mathrm{f}\left( x ight) ight)+c$, where $c$ is an arbitrary unknown constant and $\mathrm{f}\left( x ight)$ is an expression to be determined in non-trigonometric form. [4]

Accepted Answer

(a) $\mathrm e^{-a}+\mathrm e^a-2$ (b) $f(x)=-\dfrac{x}{1+x^2}$

2021 ASRJC P1 Q6

2021 ASRJC P1 Q6

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