Find the general anti derivative for each function

Question

a. y=-csc^2 (3x)

b. g(x)= -3 / 1+x^2

c. y=-sec^2(8x)

d. y=e^-10x

e. f(x)=x^3 suaqre root of 7-6x^2

Answer 1

a. F(x) = cot(3x)/3 + C; b. F(x) = - 3tan^-1x + C; c. F(x) = - tan(8x)/8 + C; d. F(x) = - e^-10x/10 + C

e. F(x) = ∫x³√7 - 6x²dx = (x = √7/6sint; dx = √7/6costdt) = ∫7/6√7/6sin³t7/6cos²tdt = - 49/36√7/6∫cos²t(1 - cos²t)·d(cost).= - 49/36√7/6(cos³t/3 - cos⁵t/5) + C where cost = (1 - 6/7x²)^1/2