" 8."(1x^(2)+mx+n)/(sqrt(x))

int_(0)^(1)((3x^(2))/(sqrt(1-x^(6)))+(4x^(3))/(sqrt(1-x^(8)))+(5x^(4))/(sqrt(1-x^(10)))+...." to "n" terms ")dx=

y=Tan^(-1)((sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2)))) then find (8ddy)/(8ddx)

lim_(x rarr1)(sqrt(x^(2)+8)-sqrt(10-x^(2)))/(sqrt(x^(2)+3)-sqrt(5-x^(2)))=

int_( then the value n+lambda is )^( If )(dx)/(sqrt(x^(2)+4x+3))dx=(lx^(2)+mx+n)sqrt(x^(2)+4x+3)+lambda int(dx)/(sqrt(x^(2)+4x+3))

(3+sqrt(8))^(x^(2)-2x+1)+(3-sqrt(8))^(x^(2)-2x-1)=((2)/(3-sqrt(8))) then the difference between the greatest value and the least possible value of x is:

If I_(n)=int(x^(n)dx)/(sqrt(x^(2)+a)) then prove that I_(n)+(n-1)/(n)al_(n-2)=(1)/(n)x^(n-1)*sqrt(x^(2)+a)

Let us consider the integral of the following forms f(x_(1), sqrt(mx^(2)+nx+p))^(1//2) Case I If m gt 0 , then put sqrt(mx^(2)+nx+C)=u pm x sqrt(m) Case II If p gt 0 , then put sqrt(mx^(2)+nx+C)=u x pm sqrt(p) Case III If quadratic equation mx^(2)+nx+p=0 has real roots alpha and beta there put sqrt(mx^(2)+nx+p)=(x-alpha) u or (x-beta)u int ((x+sqrt(1+x^(2)))^(15))/(sqrt(1+x^(2))) dx is equal to

Let us consider the integral of the following forms f(x_(1), sqrt(mx^(2)+nx+p))^(1//2) Case I If m gt 0 , then put sqrt(mx^(2)+nx+C)=u pm x sqrt(m) Case II If p gt 0 , then put sqrt(mx^(2)+nx+C)=u x pm sqrt(p) Case III If quadratic equation mx^(2)+nx+p=0 has real roots alpha and beta there put sqrt(mx^(2)+nx+p)=(x-alpha) u or (x-beta)u int ((x+sqrt(1+x^(2)))^(15))/(sqrt(1+x^(2))) dx is equal to

(1)/(sqrt(x^(2)-8x-20))

Integrate the following functions w.r.t. x: (i) sin mx (ii) 2xsin(x^2+1) (iii) (tan^4sqrt(x)sec^2sqrt(x))/(sqrt(x)) (iv) (sin(tan^(-1)x)/(1+x^2)