The value of the integral _0^ 5 e ^x e ^x-1 e ^x+3 d x= - Vidyadip

MCQ

The value of the integral $\int_0^{\log 5} \frac{ e ^x \sqrt{ e ^x-1}}{ e ^x+3} d x=$

A
$3+2 \pi$
✓
$4-\pi$
C
$2+\pi$
D
$4+\pi$

✓

Answer

Correct option: B.

$4-\pi$

(B)
Put $e ^x-1= t ^2 \Rightarrow e ^x d x=2 t dt$
When $x=0, t =0$ and when $x=\log 5, t =2$
$\therefore \int_0^{\log 5} \frac{ e ^x \sqrt{ e ^x-1}}{ e ^x+3} d x=\int_0^2 \frac{2 t ^2}{ t ^2+4} dt$
$=2 \int_0^2\left(1-\frac{4}{t^2+4}\right) d t \\ =2\left[t-4 \cdot \frac{1}{2} \tan ^{-1} \frac{t}{2}\right]_0^2 \\ =2\left(2-2 \cdot \frac{\pi}{4}\right)=4-\pi$

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