Question

Statement $-1:\underset{0}{\overset{1}{\int }}\frac{\cos x}{1+x^2}dx>\frac{\pi }{4}\cos 1$
Statement $-2:$ If $f\left(x\right)$ and $g\left(x\right)$ are continuous on $[a,b]$, then $\underset{a}{\overset{b}{\int }}f\left(x\right)g\left(x\right)dx=$
$f\left(c\right)\underset{a}{\overset{b}{\int }}g\left(x\right)dx$ for some $c\in (a,b)$

• A. Statement $1$ is true and statement $2$ is correct explanation of statement $1$
• B. Statement $1$ is true and statement $2$ is not a correct explanation of statement $1$
• C. Statement $1$ is false and statement $2$ is true
• D. Statement $1$ is true and statement $2$ is false

Answer 1

The correct option is A Statement $1$ is true and statement $2$ is correct explanation of statement $1$

Statement $-2$ is generalized mean value theorem, which is true.

Now, using it, we get
$\underset{0}{\overset{1}{\int }}\frac{\cos x}{1+x^2}dx=\cos c\underset{0}{\overset{1}{\int }}\frac{1}{1+x^2}dx=\frac{\pi }{4}\cos c$
For $c\in (0,1)$
$\cos c>\cos 1\Rightarrow \frac{\pi }{4}\cos c>\frac{\pi }{4}\cos 1\Rightarrow \underset{0}{\overset{1}{\int }}\frac{\cos x}{1+x^2}dx>\frac{\pi }{4}\cos 1$
Hence, both statements are correct and second is correct explanation also.