Question Bank [2022] — Maths STD 12 Science — Question
Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]4 Marks
Question
$\int \frac{3 e ^{2 x}+5}{4 e ^{2 x}-5} d x$
✓
Answer
Let $I =\int \frac{3 e ^{2 x}+5}{4 e ^{2 x}-5} d x$
Let $3 e ^{2 x }+5= A \left(4 e ^{2 x}-5\right)+ B \frac{ d }{ d x}\left(4 e ^{2 x}-5\right)$
$ =A\left(4 e^{2 x}-5\right)+B\left(8 e^{2 x}\right)$
$\therefore 3 e^{2 x}+5=e^{2 x}(4 A+8 B)-5 A $
By equating the coefficients on both sides, we get
$4 A+8 B=3 \text { and }-5 A=5$
Solving these equations, we get
$ A=-1 \text { and } B=\frac{7}{8}$
$\therefore 3 e ^{2 x }+5=-1\left(4 e ^{2 x}-5\right)+\frac{7}{8}\left(8 e ^{2 x}\right)$
$\therefore I =\int \frac{-1\left(4 e ^{2 x}-5\right)+\frac{7}{8}\left(8 e ^{2 x}\right)}{4 e ^{2 x}-5} d x$
$=-\int d x+\frac{7}{8} \int \frac{8 e ^{2 x}}{4 e ^{2 x}-5} d x$
$\left.\therefore I =-x+\frac{7}{8} \log \left|4 e ^{2 x}-5\right|+ c \quad \ldots \ldots . . . \int \frac{ f ^{\prime}(x)}{ f (x)} d x=\log | f (x)|+ c \right]$
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