Given
[tex] \int\limits { \frac{t}{1-t^7} } \, dt [/tex]
Recall that the general geometric series representation is given by
[tex]\Sigma_0^\infty x^k= \frac{1}{1-x} [/tex]
Let [tex]x=t^7[/tex]
Thus,
[tex]\frac{1}{1-t^7}=\Sigma_0^\infty (t^7)^k=\Sigma_0^\infty t^{7k}[/tex]
Multiply both sides by t, to get
[tex]\frac{1}{1-t^7}=\Sigma_0^\infty t^{7k+1}[/tex]
Integrate both sides to get
[tex]\int { \frac{t}{1-t^7} } \, dt=\Sigma_0^\infty \frac{t^{7k+2}}{7k+2} [/tex]
