Consider the integral I =∫1 e ( cos ( ln x )+ sin (- ln x )) dx Statement-1: I=e cos (1)-1 Statement-2: ∫ limitsab(f(x)+x f prime(x)) d x=b f(b)+a f(a)

Question

Consider the integral I =∫1 e ( cos ( ln x )+ sin (- ln x )) dx Statement-1: I=e cos (1)-1 Statement-2: ∫ limitsab(f(x)+x f prime(x)) d x=b f(b)+a f(a) (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.

Tardigrade Admin · Accepted Answer

If f ( x )= cos ( ln x ), then x f prime( x )= sin (- ln x ) Statement-1 is true, statement-2 is false

Q. Consider the integral $I = \int_{1}^{e} (cos (ln x) + sin (- ln x)) d x$
Statement-1: $I = e cos (1) - 1$
Statement-2: $a \int b (f (x) + x f^{'} (x)) d x = b f (b) + a f (a)$

Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.

Statement-1 is true, statement-2 is false.

Statement-1 is false, statement-2 is true.

If $f (x) = cos (ln x)$ , then $x f^{'} (x) = sin (- ln x)$
Statement-1 is true, statement-2 is false

Q. Consider the integral I=∫1e​(cos(lnx)+sin(−lnx))dx Statement-1: I=ecos(1)−1 Statement-2: a∫b​(f(x)+xf′(x))dx=bf(b)+af(a)