Question

How do I solve this using the substitution method for finding anti derivatives?

Answer 1

A bit blurry, but if I'm making it out correctly, that's


`\displaystyle\int t√((t^2-9)^3)\,\mathrm dt=\int t(t^2-9)^(3/2)\,\mathrm dt`

Set
`u=t^2-9`. Then
`\mathrm du=2t\,\mathrm dt`, or
`t\,\mathrm dt=\frac{\mathrm du}2`. The integral is then equivalent to


`\displaystyle u^(3/2)\frac{\mathrm du}2=\frac12\int u^(3/2)\,\mathrm du`

`=\frac12(u^(5/2))/(\frac52)+C`

`=\frac12*\frac25u^(5/2)+C`

`=\frac15u^(5/2)+C`

`=\frac15(t^2-9)^(5/2)+C`