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The value of int0^(2pi)sqrt(1+sin (x/2)d...

The value of `int_0^(2pi)sqrt(1+sin (x/2)dx` is `0` b. `4` c. `2` d. `8`

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Verified by Experts

The correct answer is D. `8`
`y= int_0^2pi sqrt(sin^2(x/4)+cos^2(x/4)+2sin (x/4)cos(x/4) dx)`
`=int_0^2pi sin (x/4) + cos (x/4) dx`
`= 4(-cos (x/4) + sin (x/4)) _0^(2pi)`
`= 4[(0+1)-(-1-0)]`
`y=8`
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RD SHARMA-DEFINITE INTEGRALS-Solved Examples And Exercises
  1. int0^(pi/2) 1/(1+sinx)dx equals 0 b. 1//2 c. 2 d. 3//2

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  2. int0^oo1/(1+e^x)dx equals a. log2-1 b. log2 c. log4-1 d. log2

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  3. The value of int0^(2pi)sqrt(1+sin (x/2)dx is 0 b. 4 c. 2 d. 8

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  4. int0^(pi^2//4)(sinsqrt(x))/(sqrt(x))dx equals a. pi^2//8 b. pi//4 ...

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  5. int0^(pi//2)1/(2+cos\ x)dx equals 1/3tan^(-1)(1/(sqrt(3))) b. 2/(sqrt(...

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  6. 36/piint(pi//6)^(pi//3)(dx)/(1+sqrt(cotx)) equals to

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  7. Given that int0^oo(x^2)/((x^2+a^2)(x^2+b^2)(x^2+c^2))dx=pi/(2(a+b)(b+c...

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  8. int1^elogx\ dx= 1 b. e-1 c. e+1 d. 0

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  9. int1^(sqrt(3))1/(1+x^2)dx is equal to pi/(12) b. pi/4 c. pi/6 d. pi/3

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  10. The value of int0^(pi//2)cosx\ e^(sin x)dx\ i s 1 b. e-1 c. 0 d. -1

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  11. If int0^a1/(1+4x^2)dx=\ pi/8 , then a equals pi/2 b. pi/4 c. 1/2 d. 1

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  12. int0^1f(x)dx=1,\ int0^1xf(x)dx=a ,\ int0^1x^2f(x)dx=a^2,\ t h e n\ int...

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  13. The value of int(-pi)^pisin^3xcos^2x\ dx\ is (pi^4)/2 b. (pi^4)/4 c. ...

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  14. int(pi//6)^(pi//3)1/(sin2x)dx\ is equal to (log)e3 b. (log)"e"sqrt(3...

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  15. int(-1)^1|1-x|dx is equal to -2 b. 2 c. 0 d. 4

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  16. (lim)(n->oo){1/(2n+1)+1/(2n+2)++1/(2n+n)} I n\ (1/3) b. I n\ (2/3) c....

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  17. The value of the integral int(-2)^2|1-x^2|dx is 4 b. 2 c. -2 d. 0

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  18. int0^(pi//2)1/(1+cot^3x)dx is equal to 0 b. 1 c. pi//2 d. pi//4

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  19. int0^(pi//2)(sinx)/(sin x+cos x)dx equals to pi//2 b. pi//4 c. pi//3 d...

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  20. int0^(pi//2)xsinx\ dx is equal to pi//2 b. pi//4 c. pi d. 1

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