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Given a function g, continous everywhere such that `g (1)=5 and int _(0)^(1) g (t) dt =2.` If `f (x) =1/2 int _(0) ^(x) (x -t)^(2) g (t) dt,` then find the value of `f '(1)+f''(1).`

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VK JAISWAL ENGLISH-INDEFINITE AND DEFINITE INTEGRATION-EXERCISE (SUBJECTIVE TYPE PROBLEMS)
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  5. If {x} denotes the fractional part of x, then I = int (0) ^(100) (sqrt...

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  6. int( x^3)/(x^2-2)dx

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  7. If M be the maximum value 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. Find the vlaur of lim (n to oo) (1)/(sqrtn)(1+ (1)/(sqrt2) +(1)/(sqrt3...

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  10. The maximum value of int (-pi/2) ^((3pi)/2) sin x. f (x) dx, subject t...

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  11. Given a function 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. int sqrt (x^2+4) dx

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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 ^(3) -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|) cos...

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