A) \[\frac{4}{3}\left[ 1+{{x}^{3/4}}+\log (1+{{x}^{3/4}}) \right]+C\]
B) \[\frac{4}{3}\left[ 1+{{x}^{3/4}}-\log (1+{{x}^{3/4}}) \right]+C\]
C) \[\frac{4}{3}\left[ 1-{{x}^{3/4}}+\log (1+{{x}^{3/4}}) \right]+C\]
D) None of these
Correct Answer: B
Solution :
[b] Put \[x={{z}^{4}}\Rightarrow dx=4{{z}^{3}}dz\] \[\therefore \,\,\,\,\,\,\int{\frac{\sqrt[2]{x}}{1+\sqrt[4]{{{x}^{3}}}}dx=\int{\frac{{{z}^{2}}.4{{z}^{3}}}{1+{{z}^{3}}}dz=4\int{\frac{{{z}^{3}}.{{z}^{2}}}{{{z}^{3}}+1}dz}}}\] \[=\frac{4}{3}\int{\frac{(y-1)}{y}dy}\] \[\left[ Putting\,\,{{z}^{3}}+1=y\Rightarrow {{z}^{2}}dz=\frac{1}{3}dy \right]\] \[=\frac{4}{3}(y-log\,\,y)+C\] \[=\frac{4}{3}\left[ 1+{{x}^{3/4}}-\log (1+{{x}^{3/4}}) \right]+C\]
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