Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Integration (p-1)3 Marks
Question
Evalute : $\int \frac{2 x+1}{(x+1)(x-2)} d x$
✓
Answer
Let $I =\int \frac{2 x+1}{(x+1)(x-2)} d x$ Let $\frac{2 x+1}{(x+1)(x-2)}=\frac{A}{x+1}+\frac{B}{x-2}$ $ \therefore 2 x +1= A ( x -2)+ B ( x +1) $ Put $x+1=0$, i.e. $x=-1$, we get $ \begin{aligned} & 2(-1)+1= A (-3)+ B (0) \\ & \therefore A =\frac{1}{3} \end{aligned} $ Put $x-2=0$, i.e. $x=2$, we get $ \begin{aligned} & 2(2)+1= A (0)+ B (3) \\ & \therefore B =\frac{5}{3} \\ & \therefore \frac{2 x+1}{(x+1)(x-2)}=\frac{(1 / 3)}{x+1}+\frac{(5 / 3)}{x-2} \\ & \therefore I=\int\left[\frac{(1 / 3)}{x+1}+\frac{(5 / 3)}{x-2}\right] d x \end{aligned} $ $ \begin{aligned} & =\frac{1}{3} \int \frac{1}{x+1} d x+\frac{5}{3} \int \frac{1}{x-2} d x \\ & =\frac{1}{3} \log |x+1|+\frac{5}{3} \log |x-2|+c . \end{aligned} $
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