int(0)^(100pi)(sum(r=1)^(10)tanrx)dx is ...

`int_(0)^(100pi)(sum_(r=1)^(10)tanrx)dx` is equal to

A

`100pi`

B

`-100pi`

C

1

D

none of these

Text Solution

Verified by Experts

The correct Answer is:

D

Let
`f(x)= underset(r=1)overset(10)sum tan rx =tan x +t an 2 x+....+ tan 10x`
Cleaarly, f(x) is a periodic function with period `pi`
`:. I= underset(0)overset(100pi)int (underset(r=1)overset(10)sum tan rx) dx`
`rArr I=100 underset(0)overset(pi)int (underset(r=1)overset(10)sum tan rx) dx`
`rArr I=100 [underset(r=1)overset(10)sum underset(0)overset(pi)int "tan rx dx"]`
`:. tan r (pi-x) =- tan r x "for" r= 1,2,.., 10`
`underset(0)overset(pi)inttan rx dx=0`
Hence `I=100xxunderset(r=1)overset(10)sum0=0`

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