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Prove that int(0)^(2a)f(x)dx=int(0)^(a)[...

Prove that `int_(0)^(2a)f(x)dx=int_(0)^(a)[f(a-x)+f(a+x)]dx`

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`I=int_(0)^(a)[f(a-x)+f(a-x)]dx`
`=int_(0)^(a)f(a-x)dx+int_(0)^(a)f(a+x)dx`
`=int_(0)^(a)f(a-(a-x))dx+ubrace(int_(a)^(2a)f(t)dt)_(x+a=t)`
`-int_(0)^(a)f(x)dx+ubrace(int_(a)^(2a)f(x)dx)_("Change of dummy variable")`
`=int_(0)^(2a)f(x)dx`
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CENGAGE PUBLICATION-DEFINITE INTEGRATION -JEE ADVANCED
  1. Prove that int(0)^(2a)f(x)dx=int(0)^(a)[f(a-x)+f(a+x)]dx

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

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

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

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

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

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

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

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

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

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

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

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  13. about to only mathematics

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

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

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

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

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

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

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

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

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