A) \[2x+\alpha \]
B) \[x+\alpha \]
C) \[\frac{x}{2}+\alpha \]
D) \[{{x}^{2}}+\alpha \]
Correct Answer: C
Solution :
[c] We check from the given options one by one. Options [a] and [b] do not satisfy. We check option (c). Let \[f(x)=\frac{x}{2}+\alpha \] \[\therefore \int{\frac{dx}{\frac{x}{2}+\alpha }}=\int{\frac{2dx}{(x+2\alpha )}}\] \[=2\log (x+2\alpha )+{{c}_{1}}=\log {{(x+2\alpha )}^{2}}+{{c}_{1}}\] \[=\log {{\left( \frac{x}{2}+\alpha \right)}^{2}}+\log {{2}^{2}}+{{c}_{1}}\] \[=\log {{\left( \frac{x}{2}+\alpha \right)}^{2}}+c\]
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