Home
Class 12
MATHS
The value of int(0)^(1)(tan^(-1)((x)/(x+...

The value of `int_(0)^(1)(tan^(-1)((x)/(x+1)))/(0tan^(-1)((1+2x-2x^(2))/(2)))dx` is

A

`1//4`

B

`1//2`

C

`1`

D

`2`

Text Solution

Verified by Experts

The correct Answer is:
B
`I=int_(0)^(1)("tan"^(-1)(x/(x+1)))/("tan"^(-1)((1+2x-2x^(2))/2))dx`…………….1
`=int_(0)^(1)("tan"^(-1)((1-x)/((1-x)+1)))/("tan"^(-1)((1+2(1-x)-2(1-x)^(2))2))dx`
`=int_(0)^(1)("tan"^(-1)((1-x)/(2-x)))/("tan"^(-1)((1+2x-2x^(2))/2))dx`……….2
ADding 1 and 2 we get
`2I=int_(0)^(1)("tan"^(-1)(x/(x+1))+"tan"^(-1)((1-x)/(2-x)))/("tan"^(-1)((1+2x-2x^(2))/2))dx`
`=int_(0)^(1)("tan"^(-1)((x/(x+1)+(1-x)/(2-x))/(1-x/(x+1) . (1-x)/(2-x))))/("tan"^(-1)((1+2x-2x^(2))/2))dx`
`=int_(0)^(1)((1+2x-2x^(2))/(2))/("tan"^(-1)((1+2x-2x^(2))/2))dx=int_(0)^(1) 1dx=1`
`implies I=1//2`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise (Multiple)|27 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise (Comprehension)|31 Videos

  • DEFINITE INTEGRATION

    CENGAGE|Exercise Exercise 8.11|6 Videos

  • CURVE TRACING

    CENGAGE|Exercise Exercise|24 Videos

  • DETERMINANT

    CENGAGE|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

2int_(0)^(1)(tan^(-1)x)/(x)dx=

int_(0)^(1)(tan^(-1)x)/(1+x^(2))dx

int_(0)^(1)tan (2x-1)dx=

Evaluate: int_(0)^(1)x(tan^(-1)x)^(2)dx

Evaluate int_(0)^(1)(tan^(-1)x)/(1+x^(2))dx

Evaluate int_(0)^(1)(tan^(-1)x)/((1+x^(2)))dx

int_(0)^(1)(tan^(-1)x)^2/(1+x^(2))dx

int_(0)^(oo)(tan^(-1)x)/(1+x^(2))dx

int_(0)^(1)Tan^(-1)((2x)/(1-x^(2)))dx=

int_(0)^(1)tan^(-1)(1-x+x^(2))dx=

CENGAGE-DEFINITE INTEGRATION -Exercise (Single)
  1. The value f the integral int(-pi)^pisinm xsinn xdx , for m!=n(m , n in...

    Text Solution

    |

  2. If f(x) and g(x) are continuous functions, then int(In lamda)^(In (1//...

    Text Solution

    |

  3. The value of int(0)^(1)(tan^(-1)((x)/(x+1)))/(0tan^(-1)((1+2x-2x^(2))/...

    Text Solution

    |

  4. int(-pi/2)^(pi/2)(e^(|sinx|)cosx)/((1+e^(tanx))dxi se q u a lto e+1 ...

    Text Solution

    |

  5. The value of int(-pi)^(pi)(2x(1+sinx))/(1+cos^(2)x)dx is

    Text Solution

    |

  6. [ The value of int(-pi)^( pi)sum(r=0)^(999)cos rx(1+sum(r=1)^(999)sin ...

    Text Solution

    |

  7. Let T >0 be a fixed real number. Suppose f is continuous function such...

    Text Solution

    |

  8. int1^4(x-0. 4)dxe q u a l s(w h e r e{x}i safr a c t ion a lp a r tof(...

    Text Solution

    |

  9. The value of int0^x[cost]dt ,x in [(4n+1)pi/2,(4n+3)pi/2]a n dn in N ...

    Text Solution

    |

  10. int0^x[sint]dt ,w h e r ex in (2npi,(2n+1)pi),n in N ,a n d[dot] den...

    Text Solution

    |

  11. int(0)^(x)(2^(t))/(2^([t]))dt, where [.] denotes the greatest integer ...

    Text Solution

    |

  12. f is an odd function, It is also known that f(x) is continuous for all...

    Text Solution

    |

  13. Ifg(x)=int0^x(|sint|+|cost|)dt ,t h e ng(x+(pin)/2) is equal to, wher...

    Text Solution

    |

  14. Ifx=intc^(sint)sin^(-1)z dz ,y=intk^(sqrt(t))(sinz^2)/zdz ,t h e n(dy)...

    Text Solution

    |

  15. Let f(x)=int(2)^(x)(dt)/(sqrt(1+t^(4))) and g be the inverse of f then...

    Text Solution

    |

  16. If f(x) is differentiable and int0^(t^2)xf(x)dx=2/5t^5, then f(4/(25))...

    Text Solution

    |

  17. If f(x)=cosx-int0^x(x-t)f(t)dt ,t h e nf^(prime)(x)+f(x) is equal to -...

    Text Solution

    |

  18. A function f is continuous for all x (and not everywhere zero) such th...

    Text Solution

    |

  19. lim(x->0) 1/x [int(y->a)e^(sin^2t) dt-int(x+y->a)e^(sin^2t)dt] is equa...

    Text Solution

    |

  20. Let f(x) =int1^x e^t/tdt,x in R^+ . Then complete set of valuesof x f...

    Text Solution

    |