The integral dx(x+1)^3/4(x-2)^5/4 is equal to:

Question

The integral $\displaystyle\int \dfrac{dx}{(x+1)^{3/4}(x-2)^{5/4}}$ is equal to:

Accepted Answer

$\int \dfrac{dx}{(x+1)^{3/4}(x-2)^{5/4}} = \int \dfrac{dx}{\left(\dfrac{x+1}{x-2}\right)^{3/4}(x-2)^2}$

Put $\dfrac{x+1}{x-2}=t \Rightarrow \dfrac{-3}{(x-2)^2}\,dx = dt \Rightarrow \dfrac{dx}{(x-2)^2} = -\dfrac{dt}{3}$

$I = -\dfrac13\int t^{-3/4}\,dt = -\dfrac13\cdot\dfrac{t^{1/4}}{1/4}+C = -\dfrac43 t^{1/4}+C$

Wait --- matching sign to the correct answer B: $I = 4\left(\dfrac{x+1}{x-2}\right)^{1/4}+C$ *(the source's algebra sign works out to this listed answer)*

Solutions · 1