A) \[-4\]
B) \[-2\]
C) \[2\]
D) \[4\]
Correct Answer: C
Solution :
[c] \[f(x)=\int{\frac{({{x}^{2}}+1)}{{{({{x}^{3}}+3x+6)}^{1/3}}}}dx\] Put \[{{x}^{3}}+3x+6={{t}^{3}}\] \[\Rightarrow \,\,\,3({{x}^{2}}+1)dx=3{{t}^{2}}dt\] \[\therefore \,\,\,f(x)=\int{\frac{{{t}^{2}}\,dt}{t}}=\frac{{{t}^{2}}}{2}+C\] \[\therefore \,\,\,f(x)=\frac{1}{2}{{({{x}^{3}}+3x+6)}^{2/3}}+C\] \[f(-1)=\frac{1}{2}{{(2)}^{2/3}}+C\] \[\Rightarrow \,\,C=0\] \[\therefore \,\,f(x)=\frac{1}{2}{{({{x}^{3}}+3x+6)}^{2/3}}\] \[\therefore \,\,f(-2)=\frac{1}{2}{{(-8)}^{2/3}}=\frac{1}{2}{{[{{(-2)}^{3}}]}^{2/3}}=2\]You need to login to perform this action.
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