Q88 / 155 · WBJEE 2016 · Mathematics

$\int 2^{x}\left[f^{\prime}(x)+f(x) \log 2\right] d x$ is equal to

  • A
    $2^{x} f^{\prime}(x)+C$
  • B
    $2^{x} \log 2+C$
  • Correct answer
    C
    $2^{x} f(x)+C$
  • D
    $2^{x}+C$

Solution

Let $I=\int 2^{n}\left[f^{\prime}(x)+f(x) \log 2\right] d x$ $\begin{array}{ll}\text { Consider } & g(x)=2^{x} f(x) \\ \Rightarrow & g^{\prime}(x)=2^{x} f^{\prime}(x)+2^{x} f(x) \log 2 \\ \Rightarrow & g^{\prime}(x)=2^{x}\left[f^{\prime}(x)+f(x) \log 2\right] \\ \therefore \quad & \quad I=\int g^{\prime}(x) d x=g(x)+C=2^{x} f(x)+C\end{array}$

Hence, the correct answer is option: 'C'.

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