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Consider the integral I(m) = int(0)^(pi...

Consider the integral `I_(m) = int_(0)^(pi) (sin2mx)/(sinx ) dx`, where m is a positive integer.
What is `I_(m)` equal to ?

A

`0`

B

`1`

C

`m`

D

`2m`

Text Solution

Verified by Experts

The correct Answer is:
A
`I_(m) = int_(0)^(pi) (sin 2mx)/(sin x) dx`
`= int_(0)^(pi) (sin 2m(pi-x))/(sin (pi-x)) dx`
`= int_(0)^(pi) (sin(2mpi - 2mx))/(sin x ) dx`
`= int_(0)^(pi) (-sin2mx)/(sin x) dx`
`I_(m) = -int_(0)^(pi) (sin 2mx)/(sin x) dx`
`2I_(m) = 0rArr I_(m) = 0`.
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NDA PREVIOUS YEARS-DEFINITE INTEGRATION & ITS APPLICATION-DIRECTIONS
  1. Consider the integral I(m) = int(0)^(pi) (sin2mx)/(sinx ) dx, where ...

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  2. Consider the integral I(m) = int(0)^(pi) (sin2mx)/(sinx ) dx, where ...

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  3. Consider the integral I(m) = int(0)^(pi) (sin2mx)/(sinx ) dx, where ...

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  4. Consider the integral I(m) = int(0)^(pi) (sin2mx)/(sinx ) dx, where ...

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  5. The area of the figure formed by the lines ax + by +c = 0, ax - by + c...

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  6. The value of int(a)^(b) (x^(7) + sinx)/(cosx)dx where a + b = 0 i...

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  7. The value of integral underset(a)overset(b)int(|x|)/(x)dx, a lt b is :

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  8. int(0)^(2pi) sin^(5) (x/4) dx is equal to

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  9. int(-1)^(1)x|x|dx is equal to

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  10. The area bounded by the coordinate axes and the curve sqrt(x) + sqrt(y...

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  11. Consider the integrals A = int(0)^(pi) (sinxdx)/(sinx + cos x) and B...

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  12. Consider the integrals A = int(0)^(pi) (sinxdx)/(sinx + cos x) and B...

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  13. Consider the functions f(x) = g(x) and g(x) = [1/x] Where [.] is...

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  14. Consider the functions f(x) = g(x) and g(x) = [1/x] Where [.] is...

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  15. What is int(-2)^(2) xdx -int(-2)^(2) [x]dx equal to , where [.] ...

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  16. If int(-2)^(5) f(x) dx = 4 and int(0)^(5) {1+f(x)}dx = 7, then what is...

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  17. What is int(0)^(4pi) |cos x| dx equal to ?

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  18. What is the area bounded by the curves |y| = 1-x^(2) ?

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  19. If int(0)^(pi/2)(dx)/(3cosx + 5) = k cot^(-1) 2, then what is the v...

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  20. What is int(1)^(3) |1-x^(4)| dx equal to ?

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