A) \[\frac{4}{3}{{\left( \frac{x-1}{x+2} \right)}^{1/4}}+c\]
B) \[\frac{4}{3}{{\left( \frac{x+2}{x-1} \right)}^{1/4}}+c\]
C) \[\frac{1}{3}{{\left( \frac{x-1}{x+2} \right)}^{1/4}}+c\]
D) \[\frac{1}{3}{{\left( \frac{x+2}{x-1} \right)}^{1/4}}+c\]
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{\frac{1}{{{[{{(x-1)}^{3}}{{(x+2)}^{5}}]}^{1/4}}}}\,dx=\int_{{}}^{{}}{\frac{1}{{{\left( \frac{x-1}{x+2} \right)}^{3/4}}{{(x+2)}^{2}}}}\,dx\] \[=\frac{1}{3}\int_{{}}^{{}}{\frac{1}{{{t}^{3/4}}}\,dt}\], \[\left\{ \because \,\,\,\frac{x-1}{x+2}=t\Rightarrow \frac{3}{{{(x+2)}^{2}}}\,dx=dt \right\}\] \[=\frac{1}{3}\left( \frac{{{t}^{1/4}}}{1/4} \right)+c=\frac{4}{3}{{t}^{1/4}}+c=\frac{4}{3}{{\left( \frac{x-1}{x+2} \right)}^{1/4}}+c\].
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