Question

How do you integrate `int troot3(t-4)dt`?

Answer 1

`3/7(t-4)^(7/3)+3(t-4)^(4/3)+C`

Explanation:

`I=inttroot3(t-4)color(white).dt`

Apply the substitution `u=t-4`. This also implies that `t=u+4` and `du=dt`. Then:

`I=int(u+4)root3ucolor(white).du`

We can write `root3u` with a fractional exponent and then distribute:

`I=int(u+4)u^(1/3)color(white).du`

`I=intu^(4/3)color(white).du+4intu^(1/3)color(white).du`

Integrate both using the rule `intu^ncolor(white).du=u^(n+1)/(n+1)+C`:

`I=u^(7/3)/(7/3)+4(u^(4/3)/(4/3))+C`

`I=3/7u^(7/3)+3u^(4/3)+C`

Since `u=t-4`:

`I=3/7(t-4)^(7/3)+3(t-4)^(4/3)+C`