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Integration (p-1) — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Integration (p-1)3 Marks
Question
Evalute : $\int \frac{2 e^x+5}{2 e^x+1} d x$
✓

Answer

Let $I =\int \frac{2 e ^{ x }+5}{2 e ^{ x }+1} dx$
Let $2 e^x+5=A\left(2 e^x+1\right)+B \frac{d}{d x}\left(2 e^x+1\right)$
$
\begin{aligned}
& =2 A e^x+A+B\left(2 e^x\right) \\
& \therefore 2 e^x+5=(2 A+2 B) e^x+A
\end{aligned}
$
Comparing the coefficients of $e ^{ x }$ and constant term on both sides, we get
$
2 A +2 B =2 \text { and } A =5
$
Solving these equations, we get
$
\begin{aligned}
& B =-4 \\
& \therefore I =\int \frac{5\left(2 e ^{ x }+1\right)-4\left(2 e ^{ x }\right)}{2 e ^{ x }+1} dx \\
& =5 \int dx -4 \int \frac{2 e ^{ x }}{2 e ^{ x }+1} dx \\
& \therefore I =5 x -4 \log \left|2 e ^{ x }+1\right|+c \quad \ldots . .\left[\int \frac{ f \prime ( x )}{ f ( x )} dx =\log |f( x )|+ c \right]
\end{aligned}
$

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