5.3
Evaluate the definite integral $\int_{-1}^0(t^{\frac{1}{7}}-t^{\frac{2}{7}})dt$
Can somebody verify this solution for me? Thanks!
$\int_{-1}^0(t^{\frac{1}{7}}-t^{\frac{2}{7}})dt$
$=\frac{t^{\frac{8}{7}}}{\frac{8}{7}}-\frac{t^{\frac{9}{7}}}{\frac{9}{7}}|_{-1}^0$
$\frac{7}{8}t^{\frac{8}{7}}-\frac{7}{9}t^{\frac{9}{7}}|_{-1}^0$
$=(\frac{7}{8}(0)^{\frac{8}{7}}-\frac{7}{9}(0)^{\frac{9}{7}})-(\frac{7}{8}(-1)^{\frac{8}{7}}-\frac{7}{9}(-1)^{\frac{9}{7}})$
Note that $(-1)^{\frac{8}{7}}=(-1)^{\frac{7}{7}}(-1)^{\frac{1}{7}}=-(-1)=1$
And $(-1)^{\frac{9}{7}}=(-1)^{\frac{2}{9}}(-1)^{\frac{7}{7}}=((-1)^2)^{\frac{1}{9}}(-1)^{\frac{7}{7}}=(1)^{\frac{1}{9}}(-1)=-1$
thus we have:
$=(\frac{7}{8}(0)^{\frac{8}{7}}-\frac{7}{9}(0)^{\frac{9}{7}})-(\frac{7}{8}(-1)^{\frac{8}{7}}-\frac{7}{9}(-1)^{\frac{9}{7}})$
$=0-(\frac{7}{8}(1)-\frac{7}{9}(-1))$
$0-(\frac{7}{8}+\frac{7}{9})$
$=-\frac{7}{8}-\frac{7}{9}$
$=-\frac{63}{72}-\frac{56}{72}$
$=\frac{-119}{72}$
Perhaps it would be easier to you if you write $\sqrt[7]{t}=x$, then $t=x^7$ so $dt = 7x^6dx$ and now you have $$...=7\int_{-1}^0(x-x^2)x^6dx=...$$