Question

Assertion :Consider $I_1={\int }_0^{\frac{\pi }{4}}{e}^{x^2}dx,I_2={\int }_0^{\frac{\pi }{4}}{e}^{x}dx,I_3={\int }_0^{\frac{\pi }{4}}{e}^{x^2}\cos xdx,I_4={\int }_0^{\frac{\pi }{4}}{e}^{x^2}\sin xdx$,
then $I_2>I_1>I_3>I_4.$ Reason: For $x(0,1),x>x^2$ and $\sin x>\cos x.$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

$x>x_2^2,\forall x\in (0,\frac{\pi }{4})$ or $e^x>e^{x^2}x\in (0,\frac{\pi }{4})$
$\cos x=\sin x\forall x\in (0,\frac{\pi }{4})$
or $e^{x^2}\cos x>e^{x^2}\sin x$
or $e^x>e^{x^2}>e^{x^2}\cos x>e^{x^2}\sin x\forall x\in (0,\frac{\pi }{4})$
or $I_2>I_1>I_3>I_4$