A) \[\frac{1}{2}\,\,{{\left( \frac{x-3}{x+4} \right)}^{3/7}}+C\]
B) \[-\frac{1}{13}{{\left( \frac{x-3}{x+4} \right)}^{-13/7}}+C\]
C) \[{{\left( \frac{x-3}{x+4} \right)}^{1/7}}+C\]
D) \[-{{\left( \frac{x-3}{x+4} \right)}^{-1/7}}+C\]
Correct Answer: C
Solution :
[c] \[\int{\frac{dx}{{{(x-3)}^{\frac{6}{7}}}{{(x+4)}^{\frac{8}{7}}}}}\] \[=\int{\frac{dx}{{{\left( \frac{x-3}{x+4} \right)}^{\frac{6}{7}}}.{{(x+4)}^{2}}}}\] Let \[\frac{x-3}{x+4}=t=1-\frac{7}{x+4}\] \[=\frac{1}{7}\int{\frac{dt}{{{t}^{\frac{6}{7}}}}}\] \[dt=\frac{7}{{{(x+4)}^{2}}}dx\] \[={{t}^{\frac{1}{7}}}+C\] \[={{\left( \frac{x-3}{x+4} \right)}^{\frac{1}{7}}}+C\]
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