NDA 2019 Maths · Mathematics · Definite Integrals · hard

∫dx/x=ln |x|+C holds on any interval not containing 0. 2. ∫-1^1(dx)/(x^2=[-1/x]-1^1=-2.

Single correct option

Consider the following statements:
1.

\displaystyle\int\frac{dx}{x}=\ln|x|+C

holds on any interval not containing

0

.
2.

\displaystyle\int_{-1}^{1}\frac{dx}{x^2}=\left[-\frac1x\right]_{-1}^{1}=-2

.
Which of the statements given above is/are correct?

Show answer and solution

Answer: (A) 1 only

Statement 1 is true: for

x<0

\frac{d}{dx}\ln(-x)=\frac1x

, so

\ln|x|

works on both sides of

0

. Statement 2 is false: the integrand

\frac1{x^2}

is positive, so a negative value is impossible; the integrand is unbounded at

x=0

and the fundamental theorem cannot be applied across the discontinuity. Only statement 1 is correct.

Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake