Evaluate 2 e ^x+5 2 e ^x+1 d x - Vidyadip

Question

Evaluate $\int \frac{2 e ^x+5}{2 e ^x+1} d x$

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Answer

Let $I =\int \frac{2 e ^x+5}{2 e ^x+1} d x$
Let $2 e ^{ x }+5= A \left(2 e ^x+1\right)+ B \frac{ d }{ d x}\left(2 e ^x+1\right)$
$=2 A e^x+A+B\left(2 e^x\right)$
$\therefore 2 e ^{ x }+5=(2 A +2 B ) e ^{ x }+ A$

Comparing the coefficients of ex and constant term on both sides,
we get $2 A+2 B=2$ and $A=5$

Solving these equations, we get
$ B =-4$
$\therefore I =\int \frac{5\left(2 e ^x+1\right)-4\left(2 e ^x\right)}{2 e ^x+1} d x$
$=5 \int d x-4 \int \frac{2 e ^x}{2 e ^x+1} d x $
$\therefore I =5 x -4 \log |2 e +1|+ c \quad \ldots \ldots . .\left[\because \frac{ f ^{\prime}(x)}{ f (x)} d x=\log | f (x)|+ c \right]$

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