The integral [ 1 4 (x - 1)^3(x + 2)^5 ,dx] is equal to : (where C is a constant of integra…

Question

The integral $$\int {{1 \over {\root 4 \of {{{(x - 1)}^3}{{(x + 2)}^5}} }}} \,dx$$ is equal to : (where C is a constant of integration)

Accepted Answer

$$\int {{{dx} \over {{{(x - 1)}^{3/4}}{{(x + 2)}^{5/4}}}}} $$

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

put $${{x + 2} \over {x - 1}} = t$$

$$ =  - {1 \over 3}\int {{{dt} \over {{t^{5/4}}}}} $$

$$ = {4 \over 3}.{1 \over {{t^{1/4}}}} + C$$

$$ = {4 \over 3}{\left( {{{x - 1} \over {x + 2}}} \right)^{1/4}} + C$$

Solutions · 1