Question

How do you find the integral of `int 1/(x+5)^(1/3) dx` from 0 to 11?

Answer 1

I found `5.14`

Explanation:

I would set `x+5=t` so that `dx=dt` and solve the indefinite integral first as:
`int1/t^(1/3)dt=intt^(-1/3)dt=t^(-1/3+1)/(-1/3+1)+c=t^(^2/3)/(2/3)+c=3/2t^(2/3)+c`
but `t=x+5` so:
`=3/2(x+5)^(2/3)+c`
Now we substitute the extrema and subtract:
`=3/2(11+5)^(2/3)-3/2(0+5)^(2/3)=5.14`