A) \[\frac{{{x}^{5m}}-1}{2m{{({{x}^{2m}}+{{x}^{m}}+1)}^{2}}}\]
B) \[\frac{{{x}^{5m}}+{{x}^{4m}}}{2m{{({{x}^{2m}}+{{x}^{m}}+1)}^{2}}}\]
C) \[\frac{{{x}^{4m}}}{2m{{({{x}^{2m}}+{{x}^{m}}+1)}^{2}}}\]
D) \[\frac{2m({{x}^{5m}}-{{x}^{4m}})}{{{({{x}^{2m}}+{{x}^{m}}+1)}^{2}}}\]
Correct Answer: C
Solution :
[c] \[I=\int{\frac{{{x}^{5m-1}}+2{{x}^{4m-1}}}{{{x}^{6m}}{{(1+{{x}^{-m}}+{{x}^{-2m}})}^{3}}}}dx\] \[=\int{\frac{{{x}^{-(m+1)}}+2{{x}^{-(2m+1)}}}{{{(1+{{x}^{-m}}+{{x}^{-2m}})}^{3}}}dx}\] Putting \[(1+{{x}^{-m}}+{{x}^{-2m}})=t,\]we get \[I=\frac{-1}{m}\int{\frac{dt}{{{t}^{3}}}}=\frac{1}{2m{{t}^{2}}}+C=\frac{{{x}^{4m}}}{2m{{({{x}^{2m}}+{{x}^{m}}+1)}^{2}}}+C\]You need to login to perform this action.
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