Evaluate the integral.  `int (t^4)/(sqrt(1-t^10)) dt` Please explain each step as you solve, in detail!! thanks

Expert Answers

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Let `f(t)= t^4/sqrt(1-t^10)`

We need to find the integral of f(t).

`==gt int f(t) dt= int t^4/sqrt(1-t^10) dt`

` ==gt int f(t) dt= int t^4/sqrt((1-t^5)(1+t^5)) dt`

Let `u= t^5 ==gt du = 5t^4 dt ==gt dt= (du)/(5t^4) `

`==gt int f(t) dt= int t^4/sqrt((1-u)(1+u)) * (du)/(5t^4) `

`==gt int f(t) dt = (1/5) int (du)/sqrt(1-u^2) du`

But we know that:`int (du)/sqrt(1-u^2) = sin^-1 (u) `

==> `int f(t) dt= (1/5) sin^-1 (u) + C`

But we know that `u= t^5`

`==gt int f(t) dt = (1/5) sin^-1 (t^5) + C`

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