Home
Class 12
MATHS
If g(x)=int(0)^(x)cos^(4)t dt, then prov...

If `g(x)=int_(0)^(x)cos^(4)t dt`, then prove that `g(x+pi)=g(x)+g(pi)`.

Text Solution

Verified by Experts

We have `g(x)=int_(0)^(x)cos^(4)tdt`
`:.g(x+pi)=int_(0)^(x+pi)cos^(4)tdt`
`=int_(0)^(x)cos^(4)tdt+int_(x)^(x+pi)cos^(4)tdt`
`=g(x)+int_(0)^(pi)cos^(4)tdt [ :' "period of" cos^(4)t "is" pi]`
`=g(x)+g(pi)`
Promotional Banner

Topper's Solved these Questions

  • DEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise SOLVED EXAMPLE_TYPE|20 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

If g(x)=int_(0)^(x)cos4tdt , then g(x+pi) equals-

If g(x)= int_0^x cos^4t dt , then g(x + pi) equals

If g(x)=int_0^xcos^4tdt , then g(x+pi) equals (a) g(x)+g(pi) (b) g(x)-g(pi) (c) g(x)g(pi) (d) (g(x))/(g(pi))

If g(x)=int_(sinx)^("sin"(2x))sin^(-1)(t)dt ,t h e n : (a) g^(prime)(pi/2)=-2pi (b) g^(prime)(-pi/2)=-2pi (c) g^(prime)(-pi/2)=2pi (d) g^(prime)(pi/2)=2pi

Let f(x) be a continuous function AAx in R , except at x=0, such that int_0^a f(x)dx , ain R^+ exists. If g(x)=int_x^a(f(t))/t dt , prove that int_0^af(x)dx=int_0^ag(x)dx

If f(x)=log_ex and g(x)=e^x , prove that f{g(x)}=g{f(x)} .

If g(x)=int_0^x2|t|dt ,then (a) g(x)=x|x| (b) g(x) is monotonic (c) g(x) is differentiable at x=0 (d) gprime(x) is differentiable at x=0

f,g,h are continuous in [0,a],f(a-x)=f(x),g(a-x)=-g(x),3h(x)-4h(a-x)=5 . Then prove that int_(0)^(a)f(x)g(x)h(x)dx=0 .

If g is inverse of f then prove that f''(g(x))=-g''(x)(f'(g(x)))^(3).

If f(x)=int_0^x("cos"(sint)+"cos"(cost)dt ,t h e nf(x+pi)i s (a) f(x)+f(pi) (b) f(x)+2(pi) (c) f(x)+f(pi/2) (d) f(x)+2f(pi/2)

CENGAGE PUBLICATION-DEFINITE INTEGRATION -JEE ADVANCED
  1. If g(x)=int(0)^(x)cos^(4)t dt, then prove that g(x+pi)=g(x)+g(pi).

    Text Solution

    |

  2. Let f be a non-negative function defined on the interval [0,1]. If int...

    Text Solution

    |

  3. The value of int0^1(x^4(1-x)^4)/(1+x^2)dxi s//a r e (22)/7-pi (b) 2/...

    Text Solution

    |

  4. Let f be a real-valued function defined on the inverval (-1,1) such th...

    Text Solution

    |

  5. The value of int(in2)^sqrt(in3) (xsinx^2)/(sinx^2+sin(In6-x^2)dx is

    Text Solution

    |

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

    Text Solution

    |

  7. Let f:[1/2,1]->R (the set of all real numbers) be a positive, non-cons...

    Text Solution

    |

  8. Let f : [0,2] rarr R be a function which is continuous on [0 , 2] and ...

    Text Solution

    |

  9. int((pi)/4)^((pi)/2)(2cosecx)^17 dx

    Text Solution

    |

  10. Let f prime(x)=(192x^3)/(2+sin^4 pix) for all x in RR with f(1/2)=0. I...

    Text Solution

    |

  11. The value of int(-pi/2)^(pi/2) (x^2cosx)/(1+e^x) dx is equal to

    Text Solution

    |

  12. If In=int(-pi)^(pi) \ (sinnx)/((1+pi^x) \ sinx) \ dx, n=0,1,2,...... t...

    Text Solution

    |

  13. about to only mathematics

    Text Solution

    |

  14. Let S be the area of the region enclosed by y=e^(-x^(2)),y=0, x=0 and ...

    Text Solution

    |

  15. For a epsilonR (the set of all real numbers) a!=-1, lim(n to oo) ((1^(...

    Text Solution

    |

  16. Let f:[a,b]rarr[1,infty) be a continuous function and lt g: RrarrR be ...

    Text Solution

    |

  17. Let f:(0,oo)rarrRR be given by f(x)=oversetxunderset(1/x)inte^((t+1/...

    Text Solution

    |

  18. The option(s) with the values of a and L that satisfy the following eq...

    Text Solution

    |

  19. Let f(x)=7tan^8x+7tan^6x-3tan^4x-3tan^2x for all x in (-pi/2, pi/2). T...

    Text Solution

    |

  20. find the period of sin(x/2)-cos(x/3) is

    Text Solution

    |

  21. Let f: Rto(0,1) be a continuous function. Then, which of the following...

    Text Solution

    |