Question

integral of 1 / z square + Y square whole 3/2
​

Answer 1

$\\ \sf \int \frac{1}{( { {z}^{2} + {y}^{2} })^{ \frac{3}{2} }} \: dy$

$\sf \purple{let}$

$\sf y = z tan \theta$

$\implies \sf \frac{dy}{d \theta} = z \sec ^{2} \theta$

$\implies \sf dy = z \sec^{2}\theta \: d \theta$

$\\ \sf \implies \int \small \frac{1}{( { {z}^{2} +( {z \tan \theta) }^{2} })^{ \frac{3}{2} } } z \sec^{2}\theta \: d \theta$

$\\ \sf \implies \int \small \frac{1}{ { {z}^{ \cancel3}(1 + \tan }^{2} \theta )^{ \frac{3}{2} } } \cancel z \sec^{2}\theta \: d \theta$

$\\ \sf \implies \frac{1}{ {z}^{2} } \int \ \frac{1}{ { \sec }^{3} \theta } \sec^{2}\theta \: d \theta$

$\\ \sf \implies \frac{1}{ {z}^{2} } \int \ \frac{1}{ { \sec } \theta } \: d \theta$

$\\ \sf \implies \frac{1}{ {z}^{2} } \int \ cos \theta \: d \theta$

$\sf \implies \frac{1}{ {z}^{2} } \ sin \theta +\tt I_{c}$

$\sf \implies \frac{\ sin \theta }{ {z}^{2} } + \tt I_{c} \sf \: ...(1)$

$\sf \purple{but}$

$\sf y = z \tan \theta$

$\implies \sf \tan \theta = \frac{y}{z}$

$\implies \sf \tan ^{2} \theta = \frac{ {y}^{2} }{ {z}^{2} }$

$\implies \sf \sec ^{2} \theta - 1= \frac{ {y}^{2} }{ {z}^{2} }$

$\implies \sf \sec ^{2} \theta = 1 + \frac{ {y}^{2} }{ {z}^{2} }$

$\implies \sf \sec ^{2} \theta = \frac{ {z}^{2} + {y}^{2} }{ {z}^{2} }$

$\implies \sf \cos ^{2} \theta = \frac{ {z}^{2} }{ {z}^{2}+ {y}^{2} }$

$\implies \sf 1 - \sin ^{2} \theta = \frac{ {z}^{2} }{ {z}^{2}+ {y}^{2} }$

$\implies \sf \sin ^{2} \theta =1 - \frac{ {z}^{2} }{ {z}^{2}+ {y}^{2} }$

$\implies \sf \sin ^{2} \theta = \frac{ {z}^{2} + {y}^{2} - {z}^{2} }{ {z}^{2}+ {y}^{2} }$

$\implies \sf \sin ^{2} \theta = \frac{ {y}^{2} }{ {z}^{2}+ {y}^{2} }$

$\therefore \sf \sin \theta = \frac{ y }{ \sqrt{ {z}^{2}+ {y}^{2}} }$

$\sf \purple{substituting \: in \: (1)}$

$\sf \frac{\ sin \theta }{ {z}^{2} } + \tt I_{c} \sf$

$\sf \implies \frac{ \frac{y}{ \sqrt{ {z}^{2} + {y}^{2} } } }{ {z}^{2} } + \tt I_{c} \sf$

$\large{ \orange{ \sf \therefore \frac{ y }{ {z}^{2} \sqrt{ {z}^{2} + {y}^{2} } } + \tt I_{c} \sf }}$

HOPE IT HELPS!!