Question

Evaluate the following integrals :
$\displaystyle \sf \int \sqrt{x} \: tan \bigg \{2 {tan}^{ - 1} \bigg( \dfrac{ \sqrt{ \sqrt{1 + \sqrt{x}} + 1} \: - \sqrt{ \sqrt{1 + \sqrt{x} } - 1 } }{ \sqrt{ \sqrt{1 + \sqrt{x}} + 1} \: + \sqrt{ \sqrt{1 + \sqrt{x} } - 1 }} \bigg) \bigg \} \: d∫x$​

Answer 1

$\large{ \underline{ \underline{ \bigstar{ \: \: \: \: \: { \pmb{ \sf{Solution \: : }}}}}}}$

$\dashrightarrow \displaystyle \sf \int \sqrt{x} \tan \bigg \{2 { \tan}^{ - 1} \bigg( \dfrac{ \sqrt{ \sqrt{1 + \sqrt{x}} + 1} \: - \sqrt{ \sqrt{1 + \sqrt{x} } - 1 } }{ \sqrt{ \sqrt{1 + \sqrt{x}} + 1} \: + \sqrt{ \sqrt{1 + \sqrt{x} } - 1 }} \bigg) \bigg \} dx$

$\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{2 { \tan}^{ - 1} \bigg[ \frac{ { \bigg(\sqrt{ \sqrt{1 + \sqrt{x} } + 1 } - \sqrt{ \sqrt{1 + \sqrt{x} } - 1} \bigg)}^{2} }{ \cancel{\sqrt{1 + \sqrt{x}} }+ 1 - \cancel{\sqrt{1 + \sqrt{x} }} + 1} \bigg] \bigg \} dx$

$\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{2 { \tan}^{ - 1} \bigg[ \frac{2 \sqrt{1 + \sqrt{x} } + 2\sqrt{ \sqrt{x} } }{2} \bigg] \bigg \}dx$

$\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{ { \tan}^{ - 1} \bigg[\frac{2 \big( \sqrt{1 + \sqrt{x} } - \sqrt{ \sqrt{x} } \big)}{1 - { \big( \sqrt{1 + \sqrt{x} } - \sqrt{ \sqrt{x} } \big) }^{2} } \bigg] \bigg \} dx \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf\bigg[ \because \: \: 2 \tan^{-1} (x) = \frac{2x}{1-x²}\bigg]$

$\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{ { \tan}^{ - 1} \bigg[ \frac{2 \big( \sqrt{1 + \sqrt{x} } - \sqrt{ \sqrt{x} } \big) }{1 - 1 - \sqrt{x} - \sqrt{x} + 2 \sqrt{ \sqrt{x} + x} } \bigg] \bigg \} dx$

$\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{ { \tan}^{ - 1} \bigg[ \frac{2 \big( \sqrt{1 + \sqrt{x} } - \sqrt{ \sqrt{x} } \big) }{2 \sqrt{ \sqrt{x} } \big( \sqrt{1 + \sqrt{x} } - \sqrt{ \sqrt{x} } \big)} \bigg] \bigg \} dx$

$\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{ { \tan}^{ - 1} \bigg( \frac{1}{ \sqrt{ \sqrt{x} } } \bigg) \bigg \} dx$

$\dashrightarrow\sf \int \sqrt{x} \times \frac{1}{ \sqrt{ \sqrt{x} } } \: dx$

$\dashrightarrow\frak{\blue{\frac{4}{5} {(x)}^{ \frac{5}{4} } + c}} \: \: \: \: \: \: \: \: \: \: \: \: \: \big[ \tt{c \: \: is \: \: integral\:\: constant}\big]$

$\\ \frak \colorbox{aqua}{BriefReflexion}$

Answer 2

Answer:

$\large{ \underline{ \underline{ \bigstar{ \: \: \: \: \: { \pmb{ \sf{Solution \: : }}}}}}}★Solution:Solution:</p><p></p><p>\begin{gathered} \dashrightarrow \displaystyle \sf \int \sqrt{x} \tan \bigg \{2 { \tan}^{ - 1} \bigg( \dfrac{ \sqrt{ \sqrt{1 + \sqrt{x}} + 1} \: - \sqrt{ \sqrt{1 + \sqrt{x} } - 1 } }{ \sqrt{ \sqrt{1 + \sqrt{x}} + 1} \: + \sqrt{ \sqrt{1 + \sqrt{x} } - 1 }} \bigg) \bigg \} dx \\ \\ \end{gathered}⇢∫xtan{2tan−1(1+x+1+1+x−11+x+1−1+x−1)}dx</p><p></p><p>\begin{gathered}\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{2 { \tan}^{ - 1} \bigg[ \frac{ { \bigg(\sqrt{ \sqrt{1 + \sqrt{x} } + 1 } - \sqrt{ \sqrt{1 + \sqrt{x} } - 1} \bigg)}^{2} }{ \cancel{\sqrt{1 + \sqrt{x}} }+ 1 - \cancel{\sqrt{1 + \sqrt{x} }} + 1} \bigg] \bigg \} dx\\ \\ \end{gathered}⇢∫xtan{2tan−1[1+x+1−1+x+1(1+x+1−1+x−1)2]}dx</p><p></p><p>\begin{gathered}\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{2 { \tan}^{ - 1} \bigg[ \frac{2 \sqrt{1 + \sqrt{x} } + 2\sqrt{ \sqrt{x} } }{2} \bigg] \bigg \}dx\\ \\ \end{gathered}⇢∫xtan{2tan−1[221+x+2x]}dx</p><p></p><p>\begin{gathered}\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{ { \tan}^{ - 1} \bigg[\frac{2 \big( \sqrt{1 + \sqrt{x} } - \sqrt{ \sqrt{x} } \big)}{1 - { \big( \sqrt{1 + \sqrt{x} } - \sqrt{ \sqrt{x} } \big) }^{2} } \bigg] \bigg \} dx \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf\bigg[ \because \: \: 2 \tan^{-1} (x) = \frac{2x}{1-x²}\bigg]\\ \\ \end{gathered}⇢∫xtan{tan−1[1−(1+x−x)22(1+x−x)]}dx[∵2tan−1(x)=1−x²2x]</p><p></p><p>\begin{gathered}\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{ { \tan}^{ - 1} \bigg[ \frac{2 \big( \sqrt{1 + \sqrt{x} } - \sqrt{ \sqrt{x} } \big) }{1 - 1 - \sqrt{x} - \sqrt{x} + 2 \sqrt{ \sqrt{x} + x} } \bigg] \bigg \} dx\\ \\ \end{gathered}⇢∫xtan{tan−1[1−1−x−x+2x+x2(1+x−x)]}dx</p><p></p><p>\begin{gathered}\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{ { \tan}^{ - 1} \bigg[ \frac{2 \big( \sqrt{1 + \sqrt{x} } - \sqrt{ \sqrt{x} } \big) }{2 \sqrt{ \sqrt{x} } \big( \sqrt{1 + \sqrt{x} } - \sqrt{ \sqrt{x} } \big)} \bigg] \bigg \} dx\\ \\ \end{gathered}⇢∫xtan{tan−1[2x(1+x−x)2(1+x−x)]}dx</p><p></p><p>\begin{gathered}\dashrightarrow\sf \int \sqrt{x} \tan \bigg \{ { \tan}^{ - 1} \bigg( \frac{1}{ \sqrt{ \sqrt{x} } } \bigg) \bigg \} dx\\ \\ \end{gathered}⇢∫xtan{tan−1(x1)}dx</p><p></p><p>\begin{gathered}\dashrightarrow\sf \int \sqrt{x} \times \frac{1}{ \sqrt{ \sqrt{x} } } \: dx \\ \\ \end{gathered}⇢∫x×x1dx</p><p></p><p>\begin{gathered}\dashrightarrow\frak{\blue{\frac{4}{5} {(x)}^{ \frac{5}{4} } + c}} \: \: \: \: \: \: \: \: \: \: \: \: \: \big[ \tt{c \: \: is \: \: integral\:\: constant}\big]\\ \\ \end{gathered}⇢54(x)45+c[cisintegralconstant]</p><p></p><p>\begin{gathered} \\ \frak \colorbox{aqua}{BriefReflexion}\\ \\\end{gathered}BriefReflexion</p><p></p><p>$