The value of _ - 7 ^7 5^x 5^ [x] dx is equal to (where [.] denote… - Vidyadip

MCQ

The value of $\int\limits_{ - 7}^7 {\frac{{{5^x}}}{{{5^{[x]}}}}dx} $ is equal to (where $[.]$ denotes greatest integer function)

A
$\frac {55}{ln\ 5}$
B
$\frac {23}{ln\ 5}$
✓
$\frac {56}{ln\ 5}$
D
$0$

✓

Answer

Correct option: C.

$\frac {56}{ln\ 5}$

c
$I = \int_{ - 7}^7 {{5^{x - [x]}}} dx = \int_{ - 7}^7 {{5^{\left[ x \right]}}} dx = 14\int_0^1 {{5^x}} dx$

$=14\left(\frac{5^{x}}{\ln 5}\right)_{0}^{1}=\frac{14}{\ln 5}(5-1)=\frac{56}{\ln 5}$

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