Question

Consider the function $F(x)=\int \frac{x}{(x-1)\left(x^2+1\right)} d x$
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)

$F(x)=\frac{1}{2} \int \frac{\left(x^2+1\right)-(x-1)^2}{\left(x^2+1\right)(x-1)} d x=\frac{1}{2} \ln |x-1|-\frac{1}{2} \int \frac{x-1}{x^2+1} d x$
$=\frac{1}{2} \ln |x-1|+\frac{1}{4} \ln \left(x^2+1\right)+\frac{1}{2} \tan ^{-1} x+C$
$\therefore$ 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 $\left.x =0\right]$