Let I = \(\int\frac{dz}{[a^2+(a-z)^2]^{3/2}}\)
Put a - z = a tan θ
⇒ dz = -asec2 θ dθ
\(\therefore\) I = \(-\int\frac{asec^2\theta d\theta}{(a^2+a^2tan^2\theta)^{3/2}}\)
= \(-\int\frac{asec^2\theta d\theta}{a^3(1+tan^2\theta)^{3/2}}\)
= \(-\frac1{a^2}\int\frac{sec^2\theta d\theta}{sec^3\theta}\)
( ∵ 1 + tan2 \(\theta\) = sec2 \(\theta\))
= -\(\frac1{a^2}\int\frac1{sec\theta}d\theta\)
= - \(\frac1{a^2}\int cos\theta d\theta\)
= \(-\frac{sin\theta}{a^2}\) ( ∵ \(\int cos\theta=sin\theta\))
= \(-\frac1{a^2}\frac{tan\theta}{\sqrt{1+tan^2\theta}}\) (∵ sin \(\theta\) = \(\frac{tan\theta}{\sqrt{1+tan^2\theta}}\))
= \(-\frac1{a^2}\cfrac{\frac{a-z}a}{\sqrt{1+\frac{(a-z)^2}{a^2}}}\) \((\because tan\theta=\frac{a-z}a)\)
\(=-\frac1{a^2}\frac{a-z}{\sqrt{a^2+(a-z)^2}}\)
Hence, \(\int\frac{dz}{[a^2+(a-z)^2]^{3/2}}\) = \(-\frac1{a^2}\frac{a-z}{\sqrt{a^2+(a-z)^2}}\)