Question

Consider $I_1=\underset{0}{\overset{\pi /4}{\int }}{e}^{x^2}dx,I_2=\underset{0}{\overset{\pi /4}{\int }}{e}^{x}dx,$
$I_3=\underset{0}{\overset{\pi /4}{\int }}{e}^{x^2}\cos xdx,I_4=\underset{0}{\overset{\pi /4}{\int }}{e}^{x^2}\sin xdx$
Statement $1:I_2>I_1>I_3>I_4$
Statement $2:$ For $x\in (0,\frac{\pi }{4}),x>x^2$ and $\cos x>\sin x$

• A. Both statements $1$ and $2$ are true but statement $2$ is not the correct explanation of statement $1.$
• B. Statement $1$ is true but statement $2$ is false.
• C. Both statements $1$ and $2$ are true and statement $2$ is the correct explanation of statement $1.$
• D. Statement $1$ is false but statement $2$ is true.

Answer 1

The correct option is A Both statements $1$ and $2$ are true but statement $2$ is not the correct explanation of statement $1.$

For $x\in (0,\frac{\pi }{4})$
$x>x^2\Rightarrow e^x>e^{x^2}\Rightarrow I_2>I_1$
Also,
$1>\cos x>\sin x,\forall x\in (0,\frac{\pi }{4})\Rightarrow e^{x^2}>e^{x^2}\cos x>e^{x^2}\sin x\Rightarrow I_1>I_3>I_4\therefore I_2>I_1>I_3>I_4$