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Prove that: inta^b(f(x))/(f(x)+f(a+b-x...

Prove that: `int_a^b(f(x))/(f(x)+f(a+b-x))dx=(b-a)/2`

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True or False: underseta oversetbint (f(x))/(f(x)+f(a+b-x))dx=(a+b)/2

d/(dx)(int f(x)dx)=

int_(a)^(b)f(x)dx=F(b)-F(a) .

int_(a)^(b)f(x)dx=int_(b)^(a)f(x)dx .

int_(0)^(a)f(x)dx=int_(a)^(0)f(a-x)dx .

If g(x)=(f(x))/((x-a)(x-b)(x-c)), where f(x) is a polynomial of degree lt3, then prove that (dg(x))/(dx)=|{:(1,a,f(a)(x-a)^(-2)),(1,b,f(b)(x-b)^(-2)),(1,c,f(c)(x-c)^(-2)):} |divide|{:(a^(2),a,1),(b^(2),b,1),(c^(2),c,1):}|.

Let f be a one-one function such that f(x).f(y) + 2 = f(x) + f(y) + f(xy), AA x, y in R - {0} and f(0) = 1, f'(1) = 2 . Prove that 3(int f(x) dx) - x(f(x) + 2) is constant.

If f and g are continuous functions on [0,a] satisfying f(x)=(a-x) and g(x)+g(a-x)=2 , then show that int_(0)^(a)f(x)g(x)dx=int_(0)^(a)f(x)dx .

ARIHANT MATHS-DEFINITE INTEGRAL-Exercise (Questions Asked In Previous 13 Years Exam)
  1. Prove that: inta^b(f(x))/(f(x)+f(a+b-x))dx=(b-a)/2

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  2. The total number of distinct x in R for which |{:(x,x^(2),1+x^(3)...

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

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  4. Let f'(x)=(192x^(3))/(2+sin^(4)pix) for all x epsilonR with f(1/2)=0. ...

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  5. The option(s) with the values of aa n dL that satisfy the following eq...

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  6. Let F:RtoR be a thrice differntiable function. Suppose that F(1)=0,F(3...

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  7. Let F:RtoR be a thrice differntiable function. Suppose that F(1)=0,F(3...

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  8. If f(x)=x-[x], where [x] is the greatest integer less than or equal to...

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  9. If alpha=int0^1(e^(9x+3tan^(-1)x))((12+9x^2)/(1+x^2))dx where tan^(-1)...

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  10. The integral overset(pi//2)underset(pi//4)int (2 cosecx)^(17)dx is equ...

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  11. Let f:[0,2]vecR be a function which is continuous on [0,2] and is diff...

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  12. Match the conditions/ expressions in Column I with statement in Column...

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  13. Find x+y+z

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  14. The value of int0^1 4x^3{(d^2)/(dx^2)(1-x^2)^5}dx is

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  15. The value of the integral int(-pi//2)^(pi//2) (x^(2) + log" (pi-x)/(pi...

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  16. The value of int(In2)^(In3) (x sinx)dx is

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  17. Let f:[1,oo] be a differentiable function such that f(1)=2. If 6int1...

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  18. The value of int(0)^(1)(x^(4)(1-x)^(4))/(1+x^(2))dx is (are)

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

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  20. Let f:[0,1]toR (the set of all real numbers ) be a function. Suppose t...

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  21. Which of the following is true?

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