Consider the integrals I1=∫e1e−xcos2x dx,I2=∫01e−x2cos2x dx,I3=∫01e−x22cos2xdx,I4=∫01e−x22dxThen

Since 0 x2>x22⇒−x<−x2<−x22⇒e−x<e−x2<e−x22⇒e−xcos2x<e−xcos2x<e−x22cos2x<e−x2/2∴∫01e−xcos2xdx<∫01e−x2cos2xdx<∫01e−x22cos2xdx<∫01e−x22dx∴I1<I2<I3<I4

Consider the integrals I1=∫e1e−xcos2x dx,I2=∫01e−x2cos2x dx,I3=∫01e−x22cos2xdx,I4=∫01e−x22dxThen

Since 0 x2>x22⇒−x<−x2<−x22⇒e−x<e−x2<e−x22⇒e−xcos2x<e−xcos2x<e−x22cos2x<e−x2/2∴∫01e−xcos2xdx<∫01e−x2cos2xdx<∫01e−x22cos2xdx<∫01e−x22dx∴I1<I2<I3<I4

AI Mentor

Check Your IQ

Free Expert Demo

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

Consider the integrals $I_{1} = \int_{e}^{1} e^{- x} \cos^{2} x d x, I_{2} = \int_{0}^{1} e^{- x^{2}} \cos^{2} x d x, I_{3} = \int_{0}^{1} e^{- \frac{x^{2}}{2}} \cos^{2} x d x,$ $I_{4} = \int_{0}^{1} e^{- \frac{x^{2}}{2}} d x$
Then

see full answer

Your Exam Success, Personally Taken Care Of

1:1 expert mentors customize learning to your strength and weaknesses – so you score higher in school , IIT JEE and NEET entrance exams.

An Intiative by Sri Chaitanya

$I_{2} > I_{4} > I_{1} > I_{3}$

$I_{2} < I_{4} < I_{1} > I_{3}$

$I_{1} < I_{2} < I_{3} < I_{4}$

$I_{1} > I_{2} > I_{3} > I_{4}$

answer is C.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Since $0 < x < 1 \Rightarrow x > x^{2} > \frac{x^{2}}{2} \Rightarrow - x < - x^{2} < - \frac{x^{2}}{2} \Rightarrow e^{- x} < e^{- x^{2}} < e^{- \frac{x^{2}}{2}}$

$\Rightarrow e^{- x} \cos^{2} x < e^{- x} \cos^{2} x < e^{- \frac{x^{2}}{2}} \cos^{2} x < e^{- x^{2} / 2}$

$∴ \int_{0}^{1} e^{- x} \cos^{2} x d x < \int_{0}^{1} e^{- x^{2}} \cos^{2} x d x < \int_{0}^{1} e^{- \frac{x^{2}}{2}} \cos^{2} x d x < \int_{0}^{1} e^{- \frac{x^{2}}{2}} d x$