Practicing Success
Area bounded by $f(x) =\frac{(x-1)(x+1)}{x-2}$, x-axis and ordinates $x = 0$ and $x = 3/2$ (in square units) is |
$\frac{4}{5}$ $\frac{7}{8}$ 1 none of these |
$\frac{7}{8}$ |
Clearly, f (x) is everywhere continuous and differentiable except at $x = 2$. The signs of f (x) for different values of x are shown in Fig. A rough sketch of the curve $y = f (x)$ is as shown in Fig. Let A denote the required area. Then, $A=\int\limits_{0}^{3/2}f(x)dx$ $⇒A=\int\limits_0^1f(x)dx+\left|\int\limits_{1}^{3/2}f(x)dx\right|$ $⇒A=\int\limits_0^1\frac{(x-1)(x+1)}{x-2}dx-\int\limits_{1}^{3/2}\frac{(x-1)(x+1)}{x-2}dx$ $⇒A=\int\limits_0^1\left(x+2+\frac{3}{x-2}\right)dx-\int\limits_{1}^{3/2}\left(x+2+\frac{3}{x-2}\right)dx$ $⇒A=[\frac{(x+2)^2}{2}+3\log|x-2|]_0^1-[\frac{(x+2)^2}{2}+3\log|x-2|]_{1}^{3/2}$ $⇒A=\left(\frac{9}{2}-2-3\log 2\right)\left[\left(\frac{49}{8}+3\log \frac{1}{2}\right)-\frac{9}{2}\right]=\frac{7}{8}$ sq. units. |