Question

Evaluate the following : `int cos(root(3)(x)).dx`

Answer 1

Let I = `int cos(root(3)(x)).dx`

Put `root(3)(x)` = t
∴ x = t³
∴ dx = 3t².dt

∴ I = `int 3t^2 cos t.dt`

= `3t^2 int cos t.dt - int [d/dt (3t)^2 int cos t.dt].dt`

= `3t^2 sint - int 6t sint.dt`

= `3t^2 sint - [6t sin t.dt - int {d/dt (6t) int sin t.dt }.dt]`

= `3t^2 sint - [6t (- cos t) - int 6( - cos t).dt]`

= 3t² sin t + 6t cos t – 6  sin t + c
= 3(t² – 2) sin t + 6t cos t + c

= `3(x^(2/3) - 2) sin(root(3)(x)) + 6root(3)(x) cos(root(3)(x)) + c`.