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Messages - ruienlin

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MAT334--Lectures & Home Assignments / Re: 2.6 Q22
« on: November 22, 2018, 02:33:25 PM »
Hi Kathy, here is my solution to (a):
For Question(a),$$  \int_0^\infty \frac{d_x}{8+x^3}=\frac{1}{8} \int_0^\infty \frac{d_x}{1+\frac{x^3}{8}}$$
$$\int_0^\infty \frac{d_x}{1+\frac{x^3}{8}}=\int_0^\infty \frac{d_x}{1+ ( \frac{x}{2})^3} $$
Change variable $x/2$ to $t$ and apply (9), then it becomes
$$2\int_0^\infty \frac{d_t}{1+ t^3} = \frac{2}{3}\frac{\pi}{\sin(\frac{\pi}{3})}=\frac{4\sqrt{3}\pi}{9}$$
so,$$ \int_0^\infty \frac{d_x}{8+x^3}=\frac{\sqrt{3}\pi}{18}$$

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