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int(0)^(2pi) (xsin^(2n)x)/(sin^(2n)x+cos...

`int_(0)^(2pi) (xsin^(2n)x)/(sin^(2n)x+cos^(2n)x)dx,n gt 0`, is equal to

A

`pi`

B

`2pi`

C

`pi^(2)`

D

`(1)/(2)pi^(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
Let`I=overset(2pi) underset(0)int(xsin^(2n)x)/(sin^(2n)x+cos^(2n)x)dx" "`......(i)
Using property IV, we have
`I_(n)=overset(2pi)underset(0)int ((2pi-x)sin^(2n)(2pi-x))/(sin^(2n)(2pi-x)+cos^(2n)(2pi-x))dx`
`rArr I_(n)=overset(2pi)underset(0)int ((2pi-x)sin^(2n)x)/(sin^(2n)x+cos^(2n)x)dx`
Adding (i) and (ii), we get
`2I_(n)=2pioverset(2pi)underset(0)int (sin^(2n)x)/(sin^(2n)x+cos^(2n)x)dx`
`rArr I_(n)=overset(2pi)underset(0)int (sin^(2n)x)/(sin^(2n)x+cos^(2n)x)dx`
`rArrI_(n)=2pioverset(pi)underset(0)int (sin^(2n)x)/(sin^(2n)x+cos^(2n)x)dx " "`[Using property VII]
`rArr I_(n)=4pi overset(pi//2)underset(0)int (sin^(2n)x)/(sin^(2n)x+cos^(2n)x)dx " "`[Using property VII]
`rArrI_(n)=4pixx(pi)/(4)=pi^(2)`
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OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Chapter Test 2
  1. int(0)^(2pi) (xsin^(2n)x)/(sin^(2n)x+cos^(2n)x)dx,n gt 0, is equal to

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  2. The integral int(0)^(r pi) sin^(2x)x dx is equal to

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  3. The value of the integral int(0)^(2)x[x]dx

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  4. The value of integral sum (k=1)^(n) int (0)^(1) f(k - 1+x) dx is

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  5. Let f(x) be a funntion satifying f'(x)=f(x) with f(0)=1 and g(x) be th...

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  6. If I=int(0)^(1) cos{ 2 "cot"^(-1)sqrt((1-x)/(1+x))}dx then

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  7. The value of int(a)^(a+(pi//2))(sin^(4)x+cos^(4)x)dx is

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  8. The vaue of int(-1)^(2) (|x|)/(x)dx is

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  9. The value of int0^1 (x^(3))/(1+x^(8))dx is

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  10. The value of int(0)^(3) xsqrt(1+x)dx, is

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  11. The value of the integral int(0)^(1) log sin ((pix)/(2))dx is

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  12. The value of the integral int(0)^(pi)x log sin x dx is

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  13. If I(1)=int(0)^(oo) (dx)/(1+x^(4))dx and I(2)underset(0)overset(oo)in...

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  14. If f(x)={{:(x,"for " x lt 1),(x-1,"for " x ge1):},"then" int(0)^(2) x...

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  15. The value of the integral int(0)^(2) (1)/((x^(2)+1)^(3//2))dx is

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  16. If int(0)^(2a) f(x)dx=int(0)^(2a) f(x)dx, then

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  17. If int(0)^(36) (1)/(2x+9)dx =log k, is equal to

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  18. The value of the integral int(0)^(pi//2) sin^(6) x dx, is

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  19. If int(0)^(oo) e^(-x^(2))dx=sqrt((pi)/(2))"then"int(0)^(oo) e^(-ax^(2)...

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  20. The value of the integral int 0^oo 1/(1+x^4)dx is

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  21. If int(pi//2)^(x) sqrt(3-2sin^(2)u) dx+int(dx)^(dy) equal pi//2

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