integate3

bx =t , dx=\frac{dt}{b}

\Rightarrow I=\frac{1}{b^{2}}\int \frac{t^{2}dt}{(a+t)^{2}b}=\frac{1}{b^{3}}\int \frac{t^{2}dt}{(a+t)^{2}}

=\frac{1}{b^{3}}\int \frac{(t^{2}-a^{2}+a^{2})dt}{(a+t)^{2}}

=\frac{1}{b^{3}}\;[\int \frac{(t^{2}-a^{2})dt}{(a+t)^{2}}+\int \frac{a^{2}dt}{(a+t)^{2}}]

\Rightarrow I=\frac{1}{b^{3}}[I_{1}+I_{2}]

I₂ can be easily obtained by takina (a+t) = y , dt=dy

for I₁, \frac{(t^{2}-a^{2})}{(t+a)^{2}}=\frac{t-a}{t+a}=1-\frac{2a}{(t+a )}

hence I can be easily found out