Home
Class 12
MATHS
If 2int(0)^(1) tan^(-1)xdx=int(2)^(1)cot...

If `2int_(0)^(1) tan^(-1)xdx=int_(2)^(1)cot^(-1)(1-x+x^(2))dx`. Then`int_(0)^(1) tan^(-1)(1-x+x^(2))dx` is equal to

A

`log 2`

B

`(pi)/(2)+log2`

C

`log 4`

D

`(pi)/(2)-log4`

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
`2overset(1)underset(0)inttan^(-1)xdx=overset(1)underset(2)cot^(-1)(1-x+x^(2))dx " "`....(i)
`2overset(1)underset(0)int tan^(-1)xdx =overset(1)underset(0)int {(pi)/(2)-tan^(-1)(1-x+x^(2))}dx`
`overset(1)underset(0)inttan^(-1)(1-x+x^(2))dx=(pi)/(2)-2overset(1)underset(0)int tan^(-1)xdx`
`rArr overset(1)underset(0)int tan^(-1)(1-x+x^(2))dx=(pi)/(2)-2[x tan^(-1) x-(1)/(2)log(1+x^(2))]_(10)^(1)`
`rArr underset(0)overset(1)int tan^(-1)(1-x+x^(2))dx=(pi)/(2)-2((pi)/(4)-(1)/(2)log2)=log2`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRALS

    OBJECTIVE RD SHARMA|Exercise Section I - Solved Mcqs|145 Videos

  • DEFINITE INTEGRALS

    OBJECTIVE RD SHARMA|Exercise Section II - Assertion Reason Type|12 Videos

  • CONTINUITY AND DIFFERENTIABILITY

    OBJECTIVE RD SHARMA|Exercise Exercise|86 Videos

  • DERIVATIVE AS A RATE MEASURER

    OBJECTIVE RD SHARMA|Exercise Exercise|26 Videos

Similar Questions

Explore conceptually related problems

int_(0)^(1)x tan^(-1)xdx=

int_(0)^(1)cot^(-1)(1-x+x^(2))dx

If 2int_(0)^(1)tan^(-1)xdx=int_(0)^(1)cot^(-1)(1-x+x^(2))dx then int_(0)^(1)tan^(-1)(1-x+x^(2))dx=

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

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

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

int_(0)^(1)x^(2)tan^(-1)xdx

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

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

OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Chapter Test 2
  1. If 2int(0)^(1) tan^(-1)xdx=int(2)^(1)cot^(-1)(1-x+x^(2))dx. Thenint(0)...

    Text Solution

    |

  2. The integral int(0)^(r pi) sin^(2x)x dx is equal to

    Text Solution

    |

  3. The value of the integral int(0)^(2)x[x]dx

    Text Solution

    |

  4. The value of integral sum (k=1)^(n) int (0)^(1) f(k - 1+x) dx is

    Text Solution

    |

  5. Let f(x) be a funntion satifying f'(x)=f(x) with f(0)=1 and g(x) be th...

    Text Solution

    |

  6. If I=int(0)^(1) cos{ 2 "cot"^(-1)sqrt((1-x)/(1+x))}dx then

    Text Solution

    |

  7. The value of int(a)^(a+(pi//2))(sin^(4)x+cos^(4)x)dx is

    Text Solution

    |

  8. The vaue of int(-1)^(2) (|x|)/(x)dx is

    Text Solution

    |

  9. The value of int0^1 (x^(3))/(1+x^(8))dx is

    Text Solution

    |

  10. The value of int(0)^(3) xsqrt(1+x)dx, is

    Text Solution

    |

  11. The value of the integral int(0)^(1) log sin ((pix)/(2))dx is

    Text Solution

    |

  12. The value of the integral int(0)^(pi)x log sin x dx is

    Text Solution

    |

  13. If I(1)=int(0)^(oo) (dx)/(1+x^(4))dx and I(2)underset(0)overset(oo)in...

    Text Solution

    |

  14. If f(x)={{:(x,"for " x lt 1),(x-1,"for " x ge1):},"then" int(0)^(2) x...

    Text Solution

    |

  15. The value of the integral int(0)^(2) (1)/((x^(2)+1)^(3//2))dx is

    Text Solution

    |

  16. If int(0)^(2a) f(x)dx=int(0)^(2a) f(x)dx, then

    Text Solution

    |

  17. If int(0)^(36) (1)/(2x+9)dx =log k, is equal to

    Text Solution

    |

  18. The value of the integral int(0)^(pi//2) sin^(6) x dx, is

    Text Solution

    |

  19. If int(0)^(oo) e^(-x^(2))dx=sqrt((pi)/(2))"then"int(0)^(oo) e^(-ax^(2)...

    Text Solution

    |

  20. The value of the integral int 0^oo 1/(1+x^4)dx is

    Text Solution

    |

  21. If int(pi//2)^(x) sqrt(3-2sin^(2)u) dx+int(dx)^(dy) equal pi//2

    Text Solution

    |