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

How do I solve this using the substitution method for finding anti derivatives?-example-1
User Stuart
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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
User Asel
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