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