Home
Class 12
MATHS
If intsin^(-1)xcos^(-1)xdx=f^(-1)(x)[π/2...

If `intsin^(-1)xcos^(-1)xdx=f^(-1)(x)[π/2x-xf^(-1)(x)-2sqrt(1-x^2)]+2x+C ,t h e n`
`(a) f(x)=sinx`
`(b) f(x)=cosx`
`(c) f(x)=tanx`
(d) none of these

A

`f(x)=sinx`

B

`f(x)=cosx`

C

`A=(pi)/(4)`

D

`A=(pi)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
A, D
`int sin^(-1)x cos^(-1)x dx=int [(pi)/(2) sin^(-1)x-(sin^(-1)x)^(2)]dx`
`=(pi)/(2) (x sin^(-1)x+sqrt(1-x^(2)))-(x(sin^(-1)x)^(2)+2sin^(-1)xsqrt(1-x^(2))-2x)+C " (Integrating by parts)" `
`=sin^(-1)x[(pi)/(2)x-x sin^(-1)x-2 sqrt(1-x^(2))]+(pi)/(2)sqrt(1-x^(2))+2x+C`
` :. f^(-1)(x)=sin^(-1)x,f(x)=sinx`
Promotional Banner

Topper's Solved these Questions

  • INDEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise Exercises (Linked Comprehension Type)|17 Videos

  • INDEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise EXERCISES (Matrix Match Type)|4 Videos

  • INDEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise EXERCISES (Single Correct Answer Type)|77 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise COMPREHENSION TYPE|2 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE PUBLICATION|Exercise Single correct Answer|21 Videos

Similar Questions

Explore conceptually related problems

Let f(x)=|x-1|dot Then (a) f(x^2)=(f(x))^2 (b) f(x+y)=f(x)+f(y) (c) f(|x|)=|f(x)| (d) none of these

Ifintxlog(1+1/x)dx=f(x)log(x+1)+g(x)x^2+A x+C , then (a) f(x)=1/2x^2 (b) g(x)=logx (c) A=1 (d) none of these

If f^(prime)(x)=sqrt(2x^2-1) and y=f(x^2),t h e n(dy)/(dx) at x=1 is (a)2 (b) 1 (c) -2 (d) none of these

If int(e^(x-1))/((x^2-5x+4))2xdx=A F(x-1)+B F(x-4)+C and F(x)=int(e^x)/xdx ,t h e n

If f(x)=2x+cos^(-1) x+log (sqrt(1+x^(2))-x) , then f(x) is-

If intsinx d(secx)=f(x)-g(x)+c ,t h e n f(x)=secx (b) f(x)=tanx g(x)=2x (d) g(x)=x

ifint(dx)/(x^2(x^n+1)^(((n-1)/n)) )=-[f(x)]^(1/n)+c ,t h e nf(x)i s (a) (1+x^n) (b) 1+x^(-1) (c) x^n+x^(-n) (d) none of these

If f(x) = x+ 1/x , x>0 then the greatest value of f(x) is (a) -2 (b)0 (c) 3 (d) none of these

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

ifint(dx)/(x^2(x^n+1)^(((n-1)/n))=-[f(x)]^(1/n)+c ,t h e nf(x)i s (1+x^n) (b) 1+x^(-1) x^n+x^(-n) (d) none of these

CENGAGE PUBLICATION-INDEFINITE INTEGRATION-Exercises (Multiple Correct Answers Type)
  1. Evaluate:int(dx)/(sqrt(2e^(x)-1))=

    Text Solution

    |

  2. If intsinx d(secx)=f(x)-g(x)+c ,t h e n f(x)=secx (b) f(x)=tanx g(x...

    Text Solution

    |

  3. Evaluate: int sqrt(1+cosecx dx)

    Text Solution

    |

  4. If I=intsec^2xcos e c^4x dx=Acot^3x+Btanx+Ccotx+D , then find the valu...

    Text Solution

    |

  5. A curve g(x)=intx^(27)(1+x+x^2)^6(6x^2+5x+4)dx is passing through orig...

    Text Solution

    |

  6. If int (sqrtx)^5/((sqrtx)^7+x^6) dx= alog(x^k/(1+x^k))+c then a and k ...

    Text Solution

    |

  7. If I=int(sinx+sin^3x)/(cos2x)dx=Pcosx+Qlog|f(x)|+R , then P=1/2,Q=-3/(...

    Text Solution

    |

  8. If int(e^(x-1))/((x^2-5x+4))2xdx=A F(x-1)+B F(x-4)+C and F(x)=int(e^x)...

    Text Solution

    |

  9. If intx^2dote^(-2x)dx=e^(-2x)(a x^2+b x+c)+d , then the value of |a/(b...

    Text Solution

    |

  10. Ifint(x^4+1)/(x^6+1)dx=tan^(-1)f(x)-2/3tan^(-1)g(x)+C ,t h e n a) bo...

    Text Solution

    |

  11. If int(x^2-x+1)/((x^2+1)^(3/2))e^x dx=e^xf(x)+c, then (a) f(x) is an ...

    Text Solution

    |

  12. If int(cos4x+1)/(cotx-tanx)dx = Acos4x+B ,t h e n (a)A=-1/8 (b) B=1/...

    Text Solution

    |

  13. If intsin^(-1)xcos^(-1)xdx=f^(-1)(x)[π/2x-xf^(-1)(x)-2sqrt(1-x^2)]+2x+...

    Text Solution

    |

  14. If f(x)=int(x^8+4)/(x^4-2x^2+2)dxa n df(0)=0,t h e n f(x) is an odd f...

    Text Solution

    |

  15. If int(dx)/(x^2+a x+1)=f(g(x))+c , then f(x) is inverse trigonometric ...

    Text Solution

    |

  16. If int((1-x^7))/(x(1+x^7))dx=Plogabsx+Qlogabs(x^7+1)+c

    Text Solution

    |

  17. If int (3sinx+2cosx)/ (3cosx+2sinx) dx=a x+b ln|2sinx+3cosx|+C, then (...

    Text Solution

    |