If I1=∫0π/2 cos2⁡x1+cos2⁡xdx,I2=∫0π/2 sin2⁡x1+sin2⁡xdx I3=∫0π/2 1+2cos2⁡xsin2⁡x4+2cos2⁡xsin2⁡xdx then

I1=∫0π2 cos2⁡x1+cos2⁡xdx=∫0π2 cos2⁡π2−x1+cos2⁡π2−xdx=∫0π2 sin2⁡x1+sin2⁡xdx=I2now I1+I2=∫0π2 sin2⁡x1+sin2⁡x+cos2⁡x1+cos2⁡xdx =∫0π2 sin2⁡x+sin2⁡xcos2⁡x+cos2⁡x+sin2⁡xcos2⁡x1+sin2⁡x+cos2⁡x+sin2⁡xcos2⁡xdx=∫1+2sin2⁡xcos2⁡x2+sin2⁡xcos2⁡xdx=2I3 ⇒I2=I3

If I1=∫0π/2 cos2⁡x1+cos2⁡xdx,I2=∫0π/2 sin2⁡x1+sin2⁡xdx I3=∫0π/2 1+2cos2⁡xsin2⁡x4+2cos2⁡xsin2⁡xdx then

I1=∫0π2 cos2⁡x1+cos2⁡xdx=∫0π2 cos2⁡π2−x1+cos2⁡π2−xdx=∫0π2 sin2⁡x1+sin2⁡xdx=I2now I1+I2=∫0π2 sin2⁡x1+sin2⁡x+cos2⁡x1+cos2⁡xdx =∫0π2 sin2⁡x+sin2⁡xcos2⁡x+cos2⁡x+sin2⁡xcos2⁡x1+sin2⁡x+cos2⁡x+sin2⁡xcos2⁡xdx=∫1+2sin2⁡xcos2⁡x2+sin2⁡xcos2⁡xdx=2I3 ⇒I2=I3

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

$If I_{1} = \int_{0}^{π / 2} \frac{\cos^{2} x}{1 + \cos^{2} x} dx, I_{2} = \int_{0}^{π / 2} \frac{\sin^{2} x}{1 + \sin^{2} x} dx I_{3} = \int_{0}^{π / 2} \frac{1 + 2 \cos^{2} {xsin}^{2} x}{4 + 2 \cos^{2} {xsin}^{2} x} dx then$

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

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

$I_{1} \neq I_{2} \neq I_{3}$

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

$I_{1} = I_{2} = I_{3}$

answer is C.

(Unlock A.I Detailed Solution for FREE)

Best Courses for You

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

$\begin{aligned} I_{1} = \int_{0}^{\frac{π}{2}} \frac{\cos^{2} x}{1 + \cos^{2} x} dx \\ = \int_{0}^{\frac{π}{2}} \frac{\cos^{2} \frac{π}{2} - x}{1 + \cos^{2} \frac{π}{2} - x} dx \\ = \int_{0}^{\frac{π}{2}} \frac{\sin^{2} x}{1 + \sin^{2} x} d x = I_{2} \end{aligned}$
$n o w I_{1} + I_{2} = \int_{0}^{\frac{π}{2}} (\frac{\sin^{2} x}{1 + \sin^{2} x} + \frac{\cos^{2} x}{1 + \cos^{2} x}) d x \begin{array}{l} = \int_{0}^{\frac{π}{2}} \frac{\sin^{2} x + \sin^{2} x \cos^{2} x + \cos^{2} x + \sin^{2} x \cos^{2} x}{1 + \sin^{2} x + \cos^{2} x + \sin^{2} x \cos^{2} x} d x \\ = \int \frac{1 + 2 \sin^{2} x \cos^{2} x}{2 + \sin^{2} x \cos^{2} x} d x = 2 I_{3} \Rightarrow I_{2} = I_{3} \end{array}$