Home
Class 12
MATHS
Let f(0)=0a n dint0^2f^(prime)(2t)e^(f(2...

Let `f(0)=0a n dint_0^2f^(prime)(2t)e^(f(2t))dt=5.t h e nv a l u eoff(4)i s` log 2 (b) log 7 (c) log 11 (d) log 13

A

`log 2`

B

`log 7`

C

`log 11`

D

`log 13`

Text Solution

Verified by Experts

The correct Answer is:
C
We have `int_(0)^(2) f'(2t)e^(f(2t))dt=5`
Put `e^(f(2t))=y`
`:. 2f'(2t)e^(f(2t))dt=dy`
`:. 1/2 int_(e^(f(0)))^(e^(f(4)))dy=5`
or `int_(e^(f(0)))^(e^(f^(4))) dy =10`
or `e^(f(4))-e^(f(0))=10`
or `e^(f(4))=10+1=11`
or `f(4)=log11`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise MCQ_TYPE|27 Videos

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise LC_TYPE|31 Videos

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise CAE_TYPE|88 Videos

  • CURVE TRACING

    CENGAGE PUBLICATION|Exercise EXERCISES|24 Videos

  • DETERMINANT

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer|5 Videos

Similar Questions

Explore conceptually related problems

Let f(x)=int_0^xe^t(t-1)(t-2)dt , Then f decreases in the interval

Ifint_0^(f(x))t^2dt=xcospix ,t h e nf^(prime)(9)i s -1/9 (b) -1/3 (c) 1/3 (d) non-existent

The value of the integral int_0^oo(xlogx)/((1+x^2)^2)dx ,is (a)0 (b) log 7 (c) 5 log 13 (d) none of these

If f'(x)=f(x)+ int_(0)^(1)f(x)dx , given f(0)=1 , then find the value of f(log_(e)2) is

For all x in (0,1) (a) e^x (b) (log)_e (1+x) (c) sin x > x (d) (log)_e x > x

If int_(0)^(x)f(t)dt=x^2+int_(x)^(1)t^2f(t)dt , then f((1)/(2)) is equal to

y=f(x) satisfies the relation int_(2)^(x)f(t)dt=(x^(2))/2+int_(x)^(2)t^(2)f(t)dt The range of y=f(x) is (a) [0,∞) (b) R (c) (−∞,0] (d) [−1/2,1/2]

If x y^2=4a n d(log)_3((log)_2x)+(log)_(1/3)((log)_(1/2)y)=1,t h e nxe q u a l s 4 (b) 8 (c) 16 (d) 64

If f(x)=(log)_x(lnx),t h e nf^(prime)(x) at x=e is equal to 1/e (b) e (c) 1 (d) zero

CENGAGE PUBLICATION-DEFINITE INTEGRATION -SCQ_TYPE
  1. If the function f : [0,8] to R is differentiable, then for 0 < alpha...

    Text Solution

    |

  2. If f(x)=x^(5)+5x-1 then int(5)^(41)(dx)/((f^(-1)(x))^(5)+5f^(-1)(x)) e...

    Text Solution

    |

  3. Let f(0)=0a n dint0^2f^(prime)(2t)e^(f(2t))dt=5.t h e nv a l u eoff(4)...

    Text Solution

    |

  4. If f(x)="cos"(tan^(-1)x), then the value of the integral int0^1xf''(x)...

    Text Solution

    |

  5. The equation of the curve is y=f(x)dot The tangents at [1,f(1),[2,f(2)...

    Text Solution

    |

  6. The value of int1^e((tan^(-1)x)/x+(logx)/(1+x^2))dx ,is (a) tane (b)...

    Text Solution

    |

  7. If f(pi) = 2 and int0^pi (f(x)+f^('')(x)) sinx dx=5, then f(0) is equa...

    Text Solution

    |

  8. If int1^2e^x^2dx=a ,t h e ninte^(e^4)sqrt(1n x)dxi se q u a lto 2e^4-...

    Text Solution

    |

  9. If f(x) is continuous for all real values of x , then sum(r=1)^nf(r-1...

    Text Solution

    |

  10. T h ev a l u eofint0^(pi/2)sin|2x-alpha|dx ,w h e r ealpha in [0,pi],i...

    Text Solution

    |

  11. f(x) is a continuous function for all real values of x and satisfies i...

    Text Solution

    |

  12. If f(x)=int(-1)^(x)|t|dt, then for any xge0,f(x) equals

    Text Solution

    |

  13. The value of int(-a)^a(cos^(- 1)x-sin^(- 1)sqrt(1-x^2))dx is (a>0) w...

    Text Solution

    |

  14. The value of int1^a[x]f^(prime)(x)dx ,where a >1, and [x] denotes the ...

    Text Solution

    |

  15. int(3)^(10)[log[x]]dx is equal to (where [.] represents the greatest i...

    Text Solution

    |

  16. int(-1)^2[([x])/(1+x^2)]dx ,w h e r e[dot] denotes the greater integer...

    Text Solution

    |

  17. The value of int(-2)^1[x[1+cos((pix)/2)]+1]dx , where [.] denotes the...

    Text Solution

    |

  18. The value of int0^(2pi)[2sinx]dx ,w h e r e[dot] represents the greate...

    Text Solution

    |

  19. I1=int0^(pi/2)(sinx-cosx)/(1+sinxcosx)dx ,I2=int0^(2pi)cos^6xdx ,I3=in...

    Text Solution

    |

  20. Given int(0)^(pi//2)(dx)/(1+sinx+cosx)=A. Then the value of the defini...

    Text Solution

    |