$$ Q=qa\frac{b^{2}-a^{2}}{2}\intop_{0}^{\frac{\pi}{2}}\left(\frac{1}{\left(b^{2}+a^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\frac{1}{\left(a^{2}+b^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right)\sin\theta\,d\theta= $$ $$ -qa\frac{b^{2}-a^{2}}{2}\intop_{0}^{\frac{\pi}{2}}\left(\frac{1}{\left(b^{2}+a^{2}+2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}-\frac{1}{\left(a^{2}+b^{2}-2ab\cdot\cos(\theta)\right)^{\frac{3}{2}}}\right)\,d(\cos\theta)= $$ $$ q\frac{b^{2}-a^{2}}{2b}\left.\left(\frac{1}{\sqrt{b^{2}+a^{2}+2ab\cdot\cos(\theta)}}+\frac{1}{\sqrt{a^{2}+b^{2}-2ab\cdot\cos(\theta)}}\right)\right|_{0}^{\frac{\pi}{2}}= $$ $$ q\frac{b^{2}-a^{2}}{2b}\left(\frac{2}{\sqrt{b^{2}+a^{2}}}-\frac{1}{\sqrt{b^{2}+a^{2}+2ab}}-\frac{1}{\sqrt{a^{2}+b^{2}-2ab}}\right)= $$ $$ q\frac{b^{2}-a^{2}}{2b}\left(\frac{2}{\sqrt{b^{2}+a^{2}}}-\frac{1}{b+a}-\frac{1}{b-a}\right)= $$ $$ q\frac{b^{2}-a^{2}}{b}\left(\frac{1}{\sqrt{b^{2}+a^{2}}}-\frac{b}{b^{2}-a^{2}}\right)= $$ $$ -q\left(1-\frac{b^{2}-a^{2}}{b\sqrt{b^{2}+a^{2}}}\right). $$