Definite Integration

We know that cos‌x=1−

x2

2!

+

x4

4!

−...
So, cos‌x>1−

x2

2

for all x∈(0,1)
⇒x‌cos‌x>x−

x3

2

⇒

1

∫

0

x‌cos‌x‌d‌x>

1

∫

0

(x−

x3

2

)dx
⇒

1

∫

0

x‌cos‌x‌d‌x>

3

8

(Option A is correct)
∵x2‌cos‌x<x for all x∈(0,1)
⇒

1

∫

0

x2‌cos‌x‌d‌x<

1

∫

0

xdx
⇒

1

∫

0

x2‌cos‌x‌d‌x<

1

2

(Option C is incorrect)
Again we know that, sin‌x=x−

x3

3!

+

x5

5!

−....
So, sin‌x>x−

x3

6

for all x∈(0,1)
⇒x‌sin‌x>x2−

x4

6

1

∫

0

x‌sin‌x‌d‌x>

1

∫

0

(x2−

x4

6

)dx
⇒

1

∫

0

x‌sin‌d‌x>

3

10

(Option B is correct)
∵x2‌sin‌x>x3−

x5

6

⇒

1

∫

0

x2‌sin‌x‌d‌x>

2

9

(Option D is correct)