[" in "2sin5(5)/(105)=20(4)/(0)pi-],[quad (1+x^(2))(dy)/(dx)+y=e^(tan^(-1)x)]

e^(-x) (dy)/(dx) = y(1+ tanx + tan^(2) x)

If y = f(x) and x = g(y), where g is the inverse of f, i.e., g = f^(-1) and if (dy)/(dx) and (dx)/(dy) both exist and (dx)/(dy) ne 0 , show that (dy)/(dx) = (1)/((dx//dy)) . Hence, (1) find (d)/(dx) (tan^(-1)x) (2) If y=sin^(-1)x, -1lexle1, -(pi)/(2)leyle(pi)/(2) , then show that (dy)/(dx)=(1)/(sqrt(1-x^(2))) where |x| lt 1 .

The value of the definite integral (1)/(pi)int_((pi)/(2))^((5 pi)/(2))(e^(tan^(-1)(sin x)))/(e^(tan^(-1)(sin x)+e^(tan^(-1)(cos x)))dx is )

If (sin x)^2 =x+y find (dy)/(dx) Find (dy)/(dx) if y=sin^(-1)[2^(x+1)/(1+4^x)]

Find (dy)/(dx) if y=sin^(-1)[(6x-4sqrt(1-4x^(2)))/(5)]

Find (dy)/(dx) if y=tan^(-1)((4x)/(1+5x^(2)))+tan^(-1)((2+3x)/(3-2x))

Find (dy)/(dx) for y=tan^(-1){(1-cos x)/(sin x)},-pi

[x sin^(2)((y)/(x)) - y] dx + x dy = 0, y = (pi)/(4) when x = 1