Question

Consider the function $F(x)=\int \frac{x}{(x-1)\left(x^2+1\right)}dx$
Statement-1:$F(x)$ is discontinuous at $x=1$.
Statement-2: Integrand of $F(x)$ is discontinuous at $x=1$.

• A. Statement-1 is true, statement- 2 is true and statement- 2 is correct explanation for statement- 1 .
• B. Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
• C. Statement-1 is true, statement- 2 is false.
• D. Statement- 1 is false, statement- 2 is true.

Answer 1

Answer: (B)

$F(x)=\frac{1}{2}\int \frac{\left(x^2+1\right)-(x-1{)}^2}{\left(x^2+1\right)(x-1)}dx=\frac{1}{2}\ln |x-1|-\frac{1}{2}\int \frac{x-1}{x^2+1}dx$
$=\frac{1}{2}\ln |x-1|+\frac{1}{4}\ln \left(x^2+1\right)+\frac{1}{2}{\tan }^{-1}x+C$
$\therefore$ discontinuous at $x=1$
note that $f(x)=\int \frac{dx}{x^{1/3}}=\frac{3}{2}{x}^{2/3}+C$ is continuous although $\frac{1}{x^{1/3}}$ is discontinuous at $x=0]$