Question

$\huge{don't \: spam}$
​

Answer 1

$\mathfrak{\bf{\underline{\underline{QUESTION</p><p> \ }}}}$

$\impliesㅤㅤㅤㅤ \int \frac{ \sqrt{(1 + {x}^{2} )^{5} } }{ {x}^{6} }$

$\mathfrak{\bf{\underline{\underline{SUBSTITUTIONS \ }}}}$

$ㅤ \: ㅤx \: = \tan( \theta) \: \: \: \: \: \: \: \: \: \: \: \: \: ㅤ{ \theta} = {tan}^{ - 1} x \\ \: \: \: \: \: dx = { \sec }^{2} { \theta} \: d{ \theta}$

$\mathfrak{\bf{\underline{\underline{SOLUTION \ }}}}$

$ɪ \: = ㅤㅤ\int \frac{ \sqrt{(1 + { \tan }^{2} { \theta}}) ^{5} }{ {tan}^{6} { \theta}} . \sec ^{2} { \theta} \: d{ \theta} \\ \\ ɪ = \int \frac{ \sqrt{(sec ^{2}{ \theta}) ^{5} } }{ {tan}^{6}{ \theta} } . {sec}^{2}{ \theta} \: d{ \theta} \: \: \: \ \: \: \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: .....(1 + {tan}^{2}{ \theta} = {sec}^{2} { \theta} )$

$\: \: \: \: \: ɪ = \: \: \: \: \: \: \: \: \: \: \: \: \int \frac{ \frac{1}{ \cancel {cos}^{5}{ \theta} } }{ \frac{ {sin}^{6} { \theta}}{ \cancel{cos}{}^{{6}}{ \theta} } } . {sec}^{2} { \theta} \: d{ \theta}$

$\:ㅤ ɪ =ㅤㅤㅤㅤ \int \frac{cos{ \theta}}{ {sin}^{6} { \theta}} . {sec}^{2} { \theta} \: d{ \theta}$

$ㅤɪ =ㅤㅤㅤ \int \frac{cos{ \theta}}{ {sin}^{6}{ \theta}( {cos}^{2} { \theta})} .d{ \theta}$

$\: ɪ =ㅤㅤㅤㅤ \int \: \frac{cos{ \theta} \: d{ \theta}}{ {sin}^{6}{ \theta} - {sin}^{8}{ \theta} }$

$\mathfrak{\bf{\underline{\underline{SUBSTITUTION\ }}}}$

$ㅤㅤㅤㅤㅤㅤ \: sin{ \theta} \: = t \\ ㅤㅤㅤㅤㅤㅤㅤcos{ \theta} \: d{ \theta} = dt$

$\: ɪ =ㅤㅤㅤㅤ \int \: \frac{dt}{ {t}^{6} - {t}^{8} }$

$\: ɪ =ㅤㅤㅤㅤ\int \frac{dt}{ {t}^{6} (1 - {t}^{2} )}$

$\: ɪ =ㅤㅤㅤㅤ \int \: \frac{(1 - {t}^{2}) + {t}^{2} }{ {t}^{6}(1 - {t}^{2} )} .d{ \theta}$

$\:ɪ = ㅤㅤㅤㅤ \int \frac{1}{ {t}^{6} } + \int \frac{(1 - {t}^{2} )+ {t}^{2} }{ {t}^{4} (1 - {t}^{2} )}$

$\: ɪ = \int \frac{1}{ {t}^{6} } + \int \frac{1}{ {t}^{4} } + \int \frac{(1 - {t}^{2}) + {t}^{2} }{ {t}^{2}(1 - {t}^{2} )}$

$\: ɪ = \int \frac{1}{ {t}^{6} } + \int \frac{1}{ {t}^{4} } + \int \frac{1}{ {t}^{2} } + \int \frac{1}{(1 - {t}^{2}) }$

$\: ɪ = \int {t}^{ - 6} + \int {t}^{ - 4} + \int \frac{1}{ {t}^{2} } + \int \frac{1}{1 - {t}^{2} }$

$\: ɪ = \frac{ {t}^ {- 5} }{ - 5} + \frac{ {t}^{ - 3} }{ - 3} - \frac{1}{t} + \frac{1}{2} log | \frac{1 + t}{1 - t} | + c$

$\:ɪ = \frac{1}{2} log | \frac{1 +t }{1 -t } | - \frac{1}{t} - \frac{1}{3 {t}^{3} } - \frac{1}{5 {t}^{5} } + c$

$\: ɪ = \frac{1}{2} log | \frac{1 + sin{ \theta}}{1 - sin{ \theta}} | - \frac{1}{sin{ \theta}} - \frac{1}{3si {n}^{3} { \theta}} - \frac{1}{5si {n}^{5} { \theta}} + c$

$\: ɪ = \frac{1}{2} log | | \frac{1 + sin{ \theta}}{1 - sin{ \theta}} | - \cosec{ \theta} -3cosec ^{3} { \theta} - 5cosec ^{5} { \theta} + c$

$\: ɪ = \frac{1}{2} log | \frac{1 + sin(tan ^{ - 1} )}{ 1 - sin(tan ^{ -1} )} | - cosec( {tan}^{ - 1} ) - 3 {cosec}^{3} ( {tan}^{ - 1)} - 5cosec ^{5} (tan ^{ - 1} ) + c$