Question

$\int 2^x\left(f^{\prime }\left(x\right)+f\left(x\right)log2\right)dx$ is equal to

• A. $2^x{f}^{\prime }(x)+c$
• B. $2^{x}log2+c$
• C. $2^{x}f(x)+c$
• D. $2^x+c$

Answer 1

Answer: (C)

$\int 2^x\left(f^{\prime }\left(x\right)+f\left(x\right)log2\right)dx=I$
Let $g\left(x\right)=2^{x}f\left(x\right)$
$\Rightarrow g^{\prime }\left(x\right)=2^x{f}^{\prime }\left(x\right)+2^{x}f\left(x\right)log2$
$=2^x\left(f^{\prime }\left(x\right)+f(x)log2\right)$
$\therefore I=\int g^{\prime }\left(x\right)dx=g\left(x\right)+c=2^{x}f\left(x\right)+c$