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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 .

PRADEEP PUBLICATION-INTEGRALS-EXERCISE
  1. int (dx)/(e^x+e^(-x)) is equal to:

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  2. Fill ups: The value of underset0overset(pi/2)int (sin^nx)/(sin^nx+cos^...

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  3. Prove that: inta^b(f(x))/(f(x)+f(a+b-x))dx=(b-a)/2

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  4. Evaluate int ((x^2-3)e^x)/(x+3)^2 .

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  5. Fill ups: int xe^x/(x+1)^2dx is equal to…………….. .

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  6. Fill ups: underset0 overset(pi/2)int cosx(e^sinxdx) is equal to……….. .

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  7. Fill ups: int (sinx)/(3+4cos^2x)dx is equal to…………….

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  8. Fill ups: int x^2/(x-1)dx is equal to……….. .

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  9. Fill ups: int 1/(x^2-4x+3)dx is equal to…………… .

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  10. Fill ups: int sqrt(3-4x)dx is equal to……….. .

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  11. Fill ups: If underset0overset(pi/2)int sin^nxdx=k underset0overset(pi/...

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  12. Fill ups: underset0overset(pi/4)int (|sinx-cosx|)/(sinx-cosx)dx is equ...

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  13. True or False: Identifinite integral of a function, if its exists, is ...

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  14. True or False: int 1/sqrt(1-x^2)dx=sin^-1x and also int 1/sqrt(1-x^2)d...

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  15. True or False: int x^ndx=(x^(n+1))/(n+1)+C, where n is any integer.

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  16. True or False: int |x|dx=pmx^2/2+C

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  17. True or False: int 1/(2x)dx=1/2log|x|+C and int 1/(2x)dx=1/2log|2x|+C

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  18. True or False: underseta oversetb int f(x)dx ne underseta oversetb int...

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  19. True or False: underset(-pi/2)overset(pi/2)int sin^3xdx=0.

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  20. True or False: underseta oversetb int f(x)dx, if it exists, is a uniqu...

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