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If maximum value of `int _(0)^(1) (f (x ))^(2)` dx under the condition `-1 le f (x) le 1, int _(0)^(1) f (x) dx -0` is `(p)/(q)` (where p and q are relatively prime positive integers). Find `p+q.`

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VIKAS GUPTA (BLACK BOOK)-INDEFINITE AND DEFINITE INTEGRATION-EXERCISE (SUBJECTIVE TYPE PROBLEMS)
  1. Find the value of |a| for which the area of triangle included between ...

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  2. Let I = int (0) ^(pi) x ^(6) (pi-x) ^(8)dx, then (pi ^(15))/((""^(15) ...

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  3. If maximum value of int (0)^(1) (f (x ))^(2) dx under the condition -1...

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  4. Let a differentiable function f (x) satisfies f (x). F '(-x) . F'(x) a...

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  5. If {x} denotes the fractional part of x, then I = int (0) ^(100) (sqrt...

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  6. Let I(n) = int (0)^(pi) (sin (n + (1)/(2))x )/(sin ((x)/(2)))dx where ...

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  7. IF M be the maximum valur of 72 int (0) ^(y) sqrt(x ^(4) +(y-y^(2))^(2...

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  8. Find the number points where f (theta) = int (-1)^(1) (sin theta dx )/...

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  9. underset(nrarroo)lim[(1)/(sqrtn)+(1)/(sqrt(2n))+(1)/(sqrt(3n))+...+(1)...

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  10. The maximum value of int (-pi//2) ^(2pi//2) sin x. f (x) dx, subject t...

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  11. Given a funtion g, continous everywhere such that g (1)=5 and int (0)^...

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  12. If f (n)= 1/pi int (0) ^(pi//2) (sin ^(2) (n theta) d theta)/(sin ^(2)...

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  13. Let f (2-x) =f (2+xand f (4-x )= f (4+x). Function f (x) satisfies int...

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  14. Let l (n) =int (-1) ^(1) |x|(1+ x+ (x ^(2))/(2 ) +(x ^(2))/(3) + ........

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  15. Let lim ( x to oo) n ^((1)/(2 )(1+(1 )/(n))). (1 ^(1) . 2 ^(2) . 3 ^(3...

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  16. If int (a )^(b) |sin x |dx =8 and int (0)^(a+b) |cos x| dx=9 then the ...

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  17. If f(x),g(x),h(x) and phi(x) are polynomial in x, (int1^x f(x) h(x) dx...

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  18. If int (0)^(2)(3x ^(2) -3x +1) cos (x ^(2) -3x ^(2)+4x -2) dx = a sin ...

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  19. let f (x) = int (0) ^(x) e ^(x-y) f'(y) dy - (x ^(2) -x+1)e ^(x) Fin...

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  20. For a positive integer n, let I (n) =int (-pi)^(pi) ((pi)/(2) -|x|) co...

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