A) \[\log ({{x}^{2}}+x-2)+3\left( \frac{x-1}{x+2} \right)+c\]
B) \[2\log ({{x}^{2}}+x-2)-3\log \left( \frac{x+1}{x+2} \right)+c\]
C) \[2\log ({{x}^{2}}+x-2)-3\log \left( \frac{x-1}{x+2} \right)+c\]
D) None of the above
Correct Answer: C
Solution :
\[\int{\frac{4x-7}{{{x}^{2}}+x-2}}dx\] \[=\int{\frac{4x-7+2-2}{{{x}^{2}}+x-2}}dx\] \[=\int{\frac{2(2x+1)-9}{{{x}^{2}}+x-2}}dx\] \[=\int{\frac{2(2x+1)}{{{x}^{2}}+x-2}}dx-9\int{\frac{1}{{{x}^{2}}+x-2+\frac{1}{4}-\frac{1}{4}}}dx\] \[=2\log ({{x}^{2}}+x-2)-9\int{\frac{1}{{{\left( x+\frac{1}{2} \right)}^{2}}-{{\left( \frac{3}{2} \right)}^{2}}}}dx\] \[=2\log ({{x}^{2}}+x-2)-9.\frac{1}{2(3/2)}\log \left[ \frac{x+\frac{1}{2}-\frac{3}{2}}{x+\frac{1}{2}+\frac{3}{2}} \right]+c\] \[=2\log ({{x}^{2}}+x-2)-3\log \left[ \frac{x-1}{x+2} \right]+c\]
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