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Prove that: int0^(2pi)(xsin^(2n)x)/(s...

Prove that: `int_0^(2pi)(xsin^(2n)x)/(sin^(2n)+cos^(2n)x)dx`

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+cosx)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 ENGLISH-DEFINITE INTEGRALS-Chapter Test 2
  1. Prove that: int0^(2pi)(xsin^(2n)x)/(sin^(2n)+cos^(2n)x)dx

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

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

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  4. Let f (x) be a function satisfying f(x)=f(x) with f(0) = 1 and g be th...

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  5. If I=int(0)^(1)cos(2 cot^(-1)sqrt(((1-x)/(1+x))))dx then :

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

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

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

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

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  10. Evaluate int(0)^(1)log(sin((pix)/(2)))dx

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  11. Evaluate int(0)^(pi) xlog sinx dx

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

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  13. If f(x)={{:(x,xlt1),(x-1,xge1):}, then underset(0)overset(2)intx^(2)f(...

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  14. The value of the integral overset(1)underset(0)int (1)/((1+x^(2))^(3//...

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  15. Prove that: int0^(2a)f(x)dx=int0^(2a)f(2a-x)dxdot

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

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

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

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

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  20. The value of alpha in [0,2pi] which does not satify the equation int(p...

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  21. lim(x to 0)(int(0)^(x^(2))sinsqrt(t) dt)/(x^(3)) is equl to

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