Home
Class 12
MATHS
Show that int0^(npi+v)|sinx|dx=2n+1-cosv...

Show that `int_0^(npi+v)|sinx|dx=2n+1-cosv ,` where `n` is a positive integer and `,lt=v

Text Solution

Verified by Experts

The correct Answer is:
`2n+1-cosv`
Let `I=int_(0)^(npi+v)|sinx|dx`
`=int_(0)^(v)|sinx|dx+int_(v)^(npi+v)|sinx|dx`
`=int_(0)^(v)sinx dx+n int_(0)^(pi) |sinx+dx` [ `:'|sinx|` has period `pi`]
`=(-cosx)_(0)^(v)+n(-cosx)_(0)^(pi)`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise SCQ_TYPE|113 Videos

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise MCQ_TYPE|27 Videos

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise SOLVED EXAMPLE_TYPE|20 Videos

  • CURVE TRACING

    CENGAGE PUBLICATION|Exercise EXERCISES|24 Videos

  • DETERMINANT

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

Show that int_0^(npi+v)|sinx|dx=2n+1-cosv , where n is a positive integer and , 0<=vltpi

Show that, int_(0)^(n pi+v)|sin x|dx=2n+1-cosv , where n is a positive integer and 0 le v le pi .

Show that 9^(n+1)-8n-9 is divisible by 64, where n is a positive integer.

int (dx)/(1+nsqrt(x+1)) , n is a positive integer.

Prove that (1+i)^n+(1-i)^n=2^((n+2)/2).cos((npi)/4) , where n is a positive integer.

Using the formula of differentiation of product functions show that (d)/(dx)(x^(n))=nx^(n-1) , where n is a positive integer.

Prove that 2^n >1+nsqrt(2^(n-1)),AAn >2 where n is a positive integer.

If U_n=int_0^pi(1-cosnx)/(1-cosx)dx , where n is positive integer or zero, then show that U_(n+2)+U_n=2U_(n+1)dot Hence, deduce that int_0^(pi/2)(sin^2ntheta)/(sin^2theta)=1/2npidot

P(n):11^(n+2)+1^(2n+1) where n is a positive integer p(n) is divisible by -

Evaluate: int_0^(2pi)[sinx]dx ,where [dot] denotes the greatest integer function.

CENGAGE PUBLICATION-DEFINITE INTEGRATION -CAE_TYPE
  1. Find the value of : int0^(10)e^(2x-[2x])d(x-[x])w h e r e[dot] denote...

    Text Solution

    |

  2. If f(x) is a function satisfying f(x+a)+f(x)=0 for all x in R and pos...

    Text Solution

    |

  3. Show that int0^(npi+v)|sinx|dx=2n+1-cosv , where n is a positive integ...

    Text Solution

    |

  4. Ifint(pi/3)^xsqrt((3-sin^2t))dt+int0^ycostdt=0,then evaluate (dy)/(d...

    Text Solution

    |

  5. Iff(x)=e^(g(x))a n dg(x)=int2^x(tdt)/(1+t^4), then find the value of ...

    Text Solution

    |

  6. Evaluate (lim)(x to 4)int4^x((4t-f(t)))/((x-4))dt

    Text Solution

    |

  7. Evaluate: lim(xto0)(int(0)^(x^(2))cos^(2)tdt)/(xsinx)

    Text Solution

    |

  8. Find the points of minima for f(x)=int0^x t(t-1)(t-2)dt

    Text Solution

    |

  9. Find the equation of tangent to y=int(x^2)^(x^3)(dt)/(sqrt(1+t^2))a t...

    Text Solution

    |

  10. I f f(x)=int((pi^2)/(16))^(x^2)(sinxsinsqrt(theta))/(1+cos^2sqrt(theta...

    Text Solution

    |

  11. If f(x) is a function satisfying f(x+a)+f(x)=0 for all x in R and pos...

    Text Solution

    |

  12. Let f(x) be a differentiable function satisfying f(x)=int(0)^(x)e^((2t...

    Text Solution

    |

  13. Ifint0^1(e^t dt)/(t+1)=a ,then find the value of int(b-1)^b(e^(-t)dt)/...

    Text Solution

    |

  14. f(x)=int1^xlogt/(1+t+t^2)dt (xge1) then prove that f(x) =f(1/x)

    Text Solution

    |

  15. f(x)=int1^x(tan^(-1)(t))/t dt \ AAx in R^+, then find the value of f...

    Text Solution

    |

  16. Evaluate: int(sqrt(2)-1)^(sqrt(2)+1)((x^2-1))/((x^2+1)^2)dx

    Text Solution

    |

  17. Evaluate: int0^(e-1)((x^2+2x-1)/2)/(x+1)dx+int1^e xlogx e^(x^(2-2)/2)d...

    Text Solution

    |

  18. Find the value of int(1/2)^(2)e^(|x-1/x|)dx.

    Text Solution

    |

  19. If I(1)=int(0)^(1)(dx)/(e^(x)(1+x)) and I(2)=int(0)^(pi//4)(e^(tan^(2)...

    Text Solution

    |

  20. If IK=int1^e(1nx)^kdx(k in I^+)dx(k in I^+), then find the value of ...

    Text Solution

    |