Question

First make a substitution and then use integration by parts to evaluate the integral. integral t^11 e^-t^6 dt + C

Answer 1

It looks like you want to find

`\displaystyle \int t^(11) e^(-t^6)\,\mathrm dt`

Substitute u = -t ⁶ and du = -6t ⁵ dt. Then

`\displaystyle \int t^(11) e^(-t^6)\,\mathrm dt = \frac16 \int (-6t^5) * (-t^6) e^(-t^6)\,\mathrm dt = \frac16 \int ue^u \,\mathrm du`

Integrate by parts, taking

f = u ==> df = du

dg = eᵘ du ==> g = eᵘ

Then

`\displaystyle \frac16 \int ue^u \,\mathrm du = \frac16\left(fg-\int g\,\mathrm df\right) \\\\ =\frac16 ue^u - \frac16\int e^u\,\mathrm du \\\\ =\frac16 ue^u - \frac16 e^u + C \\\\ =-\frac16 t^6 e^(-t^6) - \frac16 e^(-t^6) + C \\\\ =\boxed{-\frac16 e^(-t^6) \left(t^6+1\right) + C}`