The integral ( (x 2 )^x+ (2 x )^x ) _2 x d x is equal to - Vidyadip

MCQ

The integral $\int\left(\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right) \log _2 x d x$ is equal to

A
$\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x+C$
✓
$\left(\frac{x}{2}\right)^x-\left(\frac{2}{x}\right)^x+C$
C
$\left(\frac{x}{2}\right)^x \log _2\left(\frac{x}{2}\right)+C$
D
$\left(\frac{ x }{2}\right)^{ x } \log _2\left(\frac{2}{ x }\right)+ C$

✓

Answer

Correct option: B.

$\left(\frac{x}{2}\right)^x-\left(\frac{2}{x}\right)^x+C$

b
$\int\left(x^x 2^{-x}+2^x x^{-x}\right) \log _2^x d x$

$\int\left(e^{x \ln x} \cdot e^{-x \ln 2}+e^{x \ln 2} \cdot e^{-x \ln x}\right) d x$

$\int\left(e^{x \ln x-x \ln 2}+e^{x \ln 2-x \ln x}\right) \frac{\ln x}{\ln 2} d x$

$\text { let } \quad x \ln x-x \ln 2=t$

$(\ln x+1-\ln 2) d x=d t$

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