Home
Class 12
MATHS
If An=int0^(pi/2)(sin(2n-1)x)/(sinx)dx ,...

If `A_n=int_0^(pi/2)(sin(2n-1)x)/(sinx)dx ,B_n=int_0^(pi/2)((sinn x)/(sinx))^2 dx` for `n inN `, Then
(A) `A_(n+1)=A_n` (B) `B_(n+1)=B_n` (C) `A_(n+1)-A_n=B_(n+1)` (D) `B_(n+1)-B_n=A_(n+1)`

A

`A_(n+1)=A_(n)`

B

`B_(n+1)=B_(n)`

C

`A_(n+1)-A_(n)=B_(n+1)`

D

`B_(n+1)-B_(n)=A_(n+1)`

Text Solution

Verified by Experts

The correct Answer is:
A, D
`A_(n+1)-A_(n)=int_(0)^(pi//2)(sin(2n+1)xsin(2n-1)x)/(sinx) dx`
`=int_(0)^(pi//2)2 cos 2nx dx=0`
or `A_(n+1)=A_(n)`
`B_(n+1)-B_(n)=int_(0)^(pi//2)(sin^(2)(n+1)x-sin^(2)nx)/(sin^(2)x)dx`
`=int_(0)^(pi//2) (sin(2n+1)x)/(sinx)dx=A_(n+1)`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise LC_TYPE|31 Videos

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise MATRIX MATCH_TYPE|6 Videos

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise SCQ_TYPE|113 Videos

  • CURVE TRACING

    CENGAGE PUBLICATION|Exercise EXERCISES|24 Videos

  • DETERMINANT

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

Let: a_n=int_0^(pi/2)(1-sint)^nsin2tdt Then find the value of lim_(n->oo)na_n

The value of int_0^pi (sin(n+1/2)x)/(sin (x/2)) dx is, n in I (a) pi/2 (b) 0 (c) pi (d) 2pi

If I_n=int_0^1(dx)/((1+x^2)^n); n in N , then prove that 2nI_(n+1)=2^(-n)+(2n-1)I_n

If int tan^(-1)((1+cos)/(sinx))dx=(pix)/(2)-(x^(2))/(n)+c , then the value of n is-

For U_n=int_0^1x^n(2-x)^n dx ; V_n=int_0^1x^n(1-x)^ndxn in N , which of the following statement(s) is/are true? (a) U_n=2^n V_n (b) U_n=2^(-n)V_n U_n=2^(2n)V_n (d) V_n=2^(-2n)U_n

If I_n=int_(-pi)^(pi) \ (sinnx)/((1+pi^x) \ sinx) \ dx, n=0,1,2,...... then which one of the following is not true ?

If alpha and beta are the rootsof he equations x^2-a x+b=0a n dA_n=alpha^n+beta^n , then which of the following is true? a) A_(n+1)=a A_n+b A_(n-1) b) A_(n+1)=b A_(n-1)+a A_n c) A_(n+1)=a A_n-b A_(n-1) d) A_(n+1)=b A_(n-1)-a A_n

Prove that int_(0)^(1)(dx)/(1+x^(n))gt1-(1)/(n)"for n"inN

Prove that: I_n=int_0^oox^(2n+1) e^ (-x^2) dx=(n !)/2,n in N .

CENGAGE PUBLICATION-DEFINITE INTEGRATION -MCQ_TYPE
  1. If int(a)^(b)|sinx|dx=8 and int(0)^(a+b)|cosx|dx=9, then find the valu...

    Text Solution

    |

  2. Iff(x)=int0^x2|t|dt ,t h e n g(x)=x|x| g(x) is monotonic g(x) is di...

    Text Solution

    |

  3. If An=int0^(pi/2)(sin(2n-1)x)/(sinx)dx ,Bn=int0^(pi/2)((sinn x)/(sinx)...

    Text Solution

    |

  4. The value of int0^oo (dx)/(1+x^4) is

    Text Solution

    |

  5. The value of int0^1e^(x^2-x)dx is (a) <1 (b) >1 (c) > e^(-1/4) (d)...

    Text Solution

    |

  6. Ifinta^b(f(x))/(f(x)+f(a+b-x))dx=10 ,t h e n (a) b=22 ,a=2 (b) b=15 ...

    Text Solution

    |

  7. The values of a for which the integral int0^2|x-a|dxgeq1 is satisfied ...

    Text Solution

    |

  8. If f(x)=int0^x |t-1| dt, where 0 lt= x lt= 2 then (a) range of f(x) ...

    Text Solution

    |

  9. If f(2-x)=f(2+x) and f(4-x)=f(4+x) for all x and f(x) is a function fo...

    Text Solution

    |

  10. If f(x)=int0^x("cos"(sint)+"cos"(cost)dt ,t h e nf(x+pi)i s (a)f(x)+f...

    Text Solution

    |

  11. If I(n)=int(0)^(pi//4) tan^(n)x dx, (ngt1 is an integer ), then (a) I(...

    Text Solution

    |

  12. If In=int0^1(dx)/((1+x^2)^n); n in N , then prove that 2nI(n+1)=2^(-n...

    Text Solution

    |

  13. L e tf:[1,oo)->Ra n df(x)=int1^x(e^t)/t dt-e^(x)dot Then

    Text Solution

    |

  14. If f(x)=inta^x[f(x)]^(-1)dx and inta^1[f(x)]^(-1)dx=sqrt(2), then

    Text Solution

    |

  15. A Function f(x) satisfies the relation f(x)=e^x+int0^1e^xf(t)dtdotT h ...

    Text Solution

    |

  16. If int0^x {int0^u f(t) dt}du is equal to (a) int0^x (x-u) f(u) (b...

    Text Solution

    |

  17. Which of the following statement(s) is/are TRUE?

    Text Solution

    |

  18. If int(0)^(x) [x] dx = int(0)^([x]) xdx, x notin integer (where, [.] a...

    Text Solution

    |

  19. Consider the function f(theta)=int(0)^(1)(|sqrt(1-x^(2))-sintheta|)/(s...

    Text Solution

    |

  20. f:[0,1)toR be a non increasing function then for alphaepsilon (0,1)

    Text Solution

    |