Let $I_1=\int\limits_0^{\pi / 4} e^{x^2} d x, I_2=\int\limits_0^{\pi / 4} e^x d x, I_3=\int\limits_0^{\pi / 4} e^{x^2} . \cos x d x, I_4=\int\limits_0^{\pi / 4} e^{x^2} . \sin x d x$ then
Answer & explanation
Correct answer: option 2
$x>x^2 ~\forall~ x \in\left(0, \frac{\pi}{4}\right)$
$\Rightarrow e^{x}>e^{x^2} ~\forall~ x \in\left(0, \frac{\pi}{4}\right)$
$\Rightarrow$ Since cos x > sin x $\forall~ \in\left(0, \frac{\pi}{4}\right)$
$\Rightarrow e^{x^2} . \cos x>e^{x^2} \sin x$
$\Rightarrow e^{x}>e^{x^2}>e^{x^2} . \cos x>e^{x^2} \sin x ~\forall~ x \in\left(0, \frac{\pi}{4}\right)$
$\Rightarrow I_2>I_1>I_3>I_4$