The value of`int_0^1 4x^3{(d^2)/(dx^2)(1-x^2)^5}dxi s`

1.	The value of`int_0^1 4x^3{(d^2)/(dx^2)(1-x^2)^5}dxi s`
Answer» Correct Answer - 2 PLAN Integration by parts `intf(x)g(x)dx = f (x) int g (x) dx - int((d)/(dx)[f(x)] intg (x) dx ) dx` Given , `I= int _(0)^(1) 4x^(3) (d^(2))/(dx^(2))(1-x^(2))^(5)dx` ` = [ 4x^(3)(d)/(dx) (1-x^(2))^(5)]_(0)^(1) - int _(0)^(1)12 x^(2) (d)/(dx) (1- x^(2))^(5)dx` `=[4x^(3)xx5(1-x^(2))^(4)(-2x)]_(0)^(1)` `-12 [ [x^(2)(1-x^(2))^(5)]_(0)^(1)- int 2x(1-x^(2))^(5)dx]` `=0-0- 12 (0-0)+12 int_(0)^(1)2x (1- x^(2))^(5)dx` `=12xx[-((1-x^(2))^(6))/(6)]_(0)^(1)=12[0+ (1)/(6)]=2`

Discussion

Consider the integral `I=int_(0)^(2pi)(dx)/(5-2cosx)` Making the substitution `"tan"1/2x=t`, we have `I=int_(0)^(2pi)(dx)/(5-2cosx)=int_(0)^(0)(2dt)/((1+t^(2))[5-2(1-t^(2))//(1+t^(2))])=0` The result is obviously wrong, since the integrand is positive and consequently the integral of this function cannot be equal to zero. Find the mistake.
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`lim_(trarr0) int_(0)^(2pi)(|sin(x+t)-sinx|)/(|t|)dx` equalsA. 2B. 4C. 43469D. 1
Given that `lim_(nrarroo) sum_(r=1)^(n) (log_(e)(n^(2)+r^(2))-2log_(e)n)/n = log_(e)2+pi/2-2`, then evaluate : ` lim_(nrarroo) (1)/(n^(2m))[(n^(2)+1^(2))^(m)(n^(2)+2^(2))^(m) "......"(2n^(2))^(m)]^(1//n)`.
`lim_(nrarroo) sum_(r=0)^(n-1) (1)/(sqrt(n^(2)-r^(2)))`
`lim_(nrarroo) (((n+1)(n+2)"........"3n)/(n^(2n)))^(1//n)` is equal to :A. `(27)/(e^(2))`B. `(9)/(e^(2))`C. `3 log3-2`D. `(18)/(e^(4))`