Home
Class 12
MATHS
Prove that pi/6 lt int0^1(dx) (sqrt(4-...

Prove that `pi/6 lt int_0^1(dx) (sqrt(4-x^2-x^3)) lt pi/(4sqrt(2))`

Text Solution

AI Generated Solution

To prove that \[ \frac{\pi}{6} < \int_0^1 \sqrt{4 - x^2 - x^3} \, dx < \frac{\pi}{4\sqrt{2}}, \] we will analyze the integral and establish the required inequalities. ...
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE ENGLISH|Exercise Jee Advanced (Single|1 Videos

  • INTRODUCTION TO VECTORS

    CENGAGE ENGLISH|Exercise MATRIX-MATCH TYPE|3 Videos

Similar Questions

Explore conceptually related problems

Prove that pi/6

Prove that pi/6leint_0^1dx/sqrt(4-x^2-x^3)lepi/(4sqrt(2))

Prove that ln 2 lt int_(0)^(1)(dx)/(sqrt(1+x^(4))) lt (pi)/2)

Prove that : int_(0)^(a) (dx)/(x+sqrt(a^(2)-x^(2)))=(pi)/(4)

Prove that 1 lt int_(0)^(2)((5-x)/(9-x^(2)))dx lt 6/5

Prove that 4 lt= int_1^3 sqrt(3+x^2) dx lt= 4sqrt(3)

Prove the following : 1 lt int_(0)^(pi//2)sqrt(sinx)dx lt sqrt(pi/2)

Prove that :int_(0)^(pi) (x)/(1 +sin^(2) x) dx =(pi^(2))/(2sqrt(2))

int_(0)^(pi//6) sqrt(1-sin 2x) dx

Prove that : int_(0)^(1) (log x)/(sqrt(1-x^(2)))dx=-(pi)/(2)log 2

CENGAGE ENGLISH-INTEGRALS-All Questions
  1. Evaluate ("lim")(n rarr oo)sum(k=1)^nk/(n^2+k^2)

    Text Solution

    |

  2. If" f, is a continuous function with int0^x f(t) dt->oo as |x|->ooth...

    Text Solution

    |

  3. Prove that pi/6 lt int0^1(dx) (sqrt(4-x^2-x^3)) lt pi/(4sqrt(2))

    Text Solution

    |

  4. Let f:[-1,2]->[0,oo) be a continuous function such that f(x)=f(1-x)fo...

    Text Solution

    |

  5. Prove that int0^x e^(x t-t^2) dt=e^(x^(2)/4)int0^x e^-(t^(2)/4)dt

    Text Solution

    |

  6. If alpha=int0^1(e^(9x+3tan^(-1)x))((12+9x^2)/(1+x^2))dx where tan^(-1)...

    Text Solution

    |

  7. Evaluate int(-4)^(-5)e^((x+5)^(2))dx+3int(1//3)^(2//3)e^(9(x-2/3)^(2))...

    Text Solution

    |

  8. Prove that for any positive integer k ,(sin2k x)/(sinx)=2[cosx+cos3x+...

    Text Solution

    |

  9. Compute the integrals: int0^oof(x^n+x^(-n))logx(dx)/x

    Text Solution

    |

  10. If for a real number y ,[y] is the greatest integral function less, th...

    Text Solution

    |

  11. Compute the integrals: int0^oof(x^n+x^(-n))logx(dx)/(1+x^2)

    Text Solution

    |

  12. int0^x{int0^uf(t)dx}\ du is equal to (a) int0^x(x-u)f(u)du (b) int0...

    Text Solution

    |

  13. Compute the integrals: int(1/e)^e1/xsin(x-1/x)dx

    Text Solution

    |

  14. STATEMENT 1: int0^pixsinxcos^2xdx=pi/2int0^pisinxcos^2x dx. STATEMEN...

    Text Solution

    |

  15. Evaluate: int0^(pi/2)xcosx dx

    Text Solution

    |

  16. Evaluate: int0^(pi/4)(sinx+cosx)/(9+16sin2x)dx

    Text Solution

    |

  17. Evaluate: int(-pi//4)^(pi//4)log(sinx+cosx)dx

    Text Solution

    |

  18. If alpha,beta(beta>alpha), are the roots of g(x)=a x^2+b x+c=0 and f...

    Text Solution

    |

  19. Consider the function f(x) satisfying the relation f(x+1)+f(x+7)=0AAx ...

    Text Solution

    |

  20. Let f: RvecR be a differentiable function having f(2)=6,f^(prime)(2)=1...

    Text Solution

    |