这是一份ed.ac.uk爱丁堡大学MATH08058作业代写的成功案例
问题 1.
\text { Evaluate } \int \frac{x}{\sqrt{\left(x^{4}+4\right)}} d x
证明 .
Put $\mathrm{x}^{2}=\mathrm{t} ; \therefore 2 \mathrm{x} d \mathrm{~d}=\mathrm{dt}$.
$$
\begin{aligned}
&\therefore \text { the given integral }=\frac{1}{2} \int \frac{1}{\sqrt{\left(t^{2}+4\right)}} d t \
&=\frac{1}{2} \sinh ^{-1}\left(\frac{t}{2}\right)=\frac{1}{2} \sinh ^{-1}\left(\frac{x^{2}}{2}\right) .
\end{aligned}
$$
MATH08058 COURSE NOTES ：
Put $x^{3}=t, \therefore 3 x^{2} d x=d t$.
$$
\begin{aligned}
&\therefore \text { the given integral }=\frac{1}{3} \int \frac{d t}{\sqrt{\left(t^{2}-9\right)}} \
&=\frac{1}{3} \cosh ^{-1}\left(\frac{t}{3}\right)=\frac{1}{3} \cosh ^{-1}\left(\frac{x^{3}}{3}\right) .
\end{aligned}
$$