Question

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

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

Answer 1

The correct option is C $2^{x}f\left(x\right)+c$

Let $I=\int 2^x\left(f^{\prime }\right(x)+f(x\left)\log 2\right)dx$

$=\int \left[2^x{f}^{\prime }\right(x)+(\ln 2\left)2^{x}f\right(x\left)\right]dx$

Substitute $u=2^{x}f\left(x\right)$

$\Rightarrow du=\left(2^x{f}^{\prime }\right(x)+f(x\left)2^x\ln 2\right)dx$

So $I=\int du=u+c=2^{x}f\left(x\right)+c$