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  4. ∫ 2x(f '(x)+f (x)log2)dx is equal to

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

WBJEEWBJEE 2016Integrals

Solution:

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