<s>[INST] Prove $\int_0^1 {3-\sqrt{5}x\over (1+\sqrt{5}x)^3} \, dx={1\over 2}$ using an alternative method Prove that
$$\int_0^1 {3-\sqrt{5}x\over (1+\sqrt{5}x)^3} \,dx={1\over 2}\tag1$$
My try:
$u=1+\sqrt{5}x$ then $du=\sqrt{5} \, dx$
$${1\over \sqrt 5}\int_1^{1+\sqrt{5}}(4u^{-3}-u^{-2}) \, du$$
$$\left. {1\over \sqrt{5}}(-2u^{-2}+u^{-1}) \right|_1^{1+\sqrt{5}}={1\over 2}$$
Prove $(1)$ using an alternative method other than substitution method.
 [/INST] $$
{3-\sqrt{5}x\over (1+\sqrt 5 \, x)^3} = \frac A {1+\sqrt 5\,x} + \frac B {(1+\sqrt 5\,x)^2} + \frac C {(1+\sqrt 5\,x)^3}
$$
Find $A$, $B$, and $C$ and then integrate each term separately.
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