$${\frac{{\mathtt{4}}{\mathtt{\,\times\,}}{pi}{\left({\mathtt{s}}\right)}}{{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{s}}\right)}^{{\mathtt{3}}}}}{\mathtt{\,\times\,}}{\mathtt{ds}}$$
What is the integral of ^^?
+128473 Let s+1 = u
Then s = u -1
And ds = 1 du
So we have
4pi∫(u - 1)/u3 du =
4pi [∫1/u-2 du - ∫ u-3 du ] + C =
4pi [ (-1)u-1 - (-1/2)u-2 ] + C =
4pi [ 1/(2u2) - 1/u ] + C =
2pi [ 1- 2u]/ u2 + C = [back substitute ....( u = s+ 1) ]
2pi [ 1 -2(s + 1) ]/ (s + 1)2 + C
2pi [ -(2s + 1) ] / (s + 1)2 + C =
-2pi (2s + 1) / (s + 1)2 + C
+128473 Let s+1 = u
Then s = u -1
And ds = 1 du
So we have
4pi∫(u - 1)/u3 du =
4pi [∫1/u-2 du - ∫ u-3 du ] + C =
4pi [ (-1)u-1 - (-1/2)u-2 ] + C =
4pi [ 1/(2u2) - 1/u ] + C =
2pi [ 1- 2u]/ u2 + C = [back substitute ....( u = s+ 1) ]
2pi [ 1 -2(s + 1) ]/ (s + 1)2 + C
2pi [ -(2s + 1) ] / (s + 1)2 + C =
-2pi (2s + 1) / (s + 1)2 + C
