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Question: Answered & Verified by Expert
$\int 2^{x}\left[f^{\prime}(x)+f(x) \log 2\right] d x$ is equal to
MathematicsIndefinite IntegrationWBJEEWBJEE 2016
Options:
  • A $2^{x} f^{\prime}(x)+C$
  • B $2^{x} \log 2+C$
  • C $2^{x} f(x)+C$
  • D $2^{x}+C$
Solution:
1293 Upvotes Verified Answer
The correct answer is: $2^{x} f(x)+C$
Let $I=\int 2^{n}\left[f^{\prime}(x)+f(x) \log 2\right] d x$
$\begin{array}{ll}\text { Consider } & g(x)=2^{x} f(x) \\ \Rightarrow & g^{\prime}(x)=2^{x} f^{\prime}(x)+2^{x} f(x) \log 2 \\ \Rightarrow & g^{\prime}(x)=2^{x}\left[f^{\prime}(x)+f(x) \log 2\right] \\ \therefore \quad & \quad I=\int g^{\prime}(x) d x=g(x)+C=2^{x} f(x)+C\end{array}$

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