Question

$\huge\green\tt\frac{ \sqrt{tanx} }{sinxcosx}$
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Answer 1

Step-by-step explanation:

Step-by-step explanation:

$\orange{\bold{\underbrace{\overbrace{❥Question᎓}}}}$

Integrate the function

$\huge\green\tt\frac{\sqrt{tanx} }{sinxcosx}$

⇛$\huge\tt\frac{ \sqrt{tanx} }{sinxcosx}$

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⇛$\huge\tt \frac{ \sqrt{tanx} }{sinxcosx \times \frac{cosx}{cosx}}$

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⇛$\huge\tt \frac{ \sqrt{tanx} }{sinx \times \frac{ {cos}^{2} x}{cosx}}$ ㅤ ㅤ ㅤ

⇛ $\huge\tt\frac{ \sqrt{tanx} }{ {cos}^{2} x \times \frac{sinx}{cosx} }$

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⇛$\huge\tt\frac{ \sqrt{tanx} }{ {cos}^{2}x \times tanx }$

⇛$\huge\tt {tan}^{ \frac{1}{2} - 1 } \times \frac{1}{ {cos}^{2} x}$ㅤ ㅤ ㅤ ㅤ ㅤ

⇛$\huge\tt {(tan)}^{ - \frac{ 1}{2} } \times \frac{1}{ {cos}^{2}x } = {(tanx)}^{ - \frac{1}{2} } \times {sec}^{2} x⇛(tan)$

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⇛$\huge\tt {(tan)}^{ - \frac{ 1}{2} } \times \frac{1}{ {cos}^{2}x } = ∫ {(tanx)}^{ - \frac{1}{2} } \times {sec}^{2} x \times dx⇛(tan)$

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$\bold\blue{☛\: Let tanx=t}$

$\bold\blue{☛ \:Differentiating \: both \: sides \: w.r.t.x}$

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⇛$\huge\tt {sec}^{2} x = \frac{dt}{dx}$

⇛$\huge\tt{dx \frac{dt}{ {sec}^{2}x }}$

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⇛$\huge\tt∴∫ {(tanx)}^{ - \frac{1}{2} } \times {sec}^{2} x \times dx$

⇛$\huge\tt ∫ {(t)}^{ - \frac{1}{2} } \times {sec}^{2} x \times \frac{dt}{ {sec}^{2}x }$

⇛$\huge\tt ∫ {t}^{ - \frac{1}{2} }$ㅤ ㅤ

⇛ $\huge\tt\frac{ {t}^{ - \frac{1}{2} + 1} }{ - \frac{1}{2} + 1 }$

⇛ $\huge\tt \frac{ {t}^{ \frac{1}{2} } }{ \frac{1}{2} } + c = 2 {t}^{ \frac{1}{2} } + c = 2 \sqrt{t}$

⇛$\huge2 \sqrt{t} + c = 2 \sqrt{tanx}$

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Answer 2

$\huge\mathcal{\mid{\mid{\underline{\pink{Good\:Morning\:}}}{\mid{\mid}}}}$

$\Large{\colorbox{pink}{\bf{\green{QuEsTiOn;-}}}}$

$\red\checkmark\:\:\bf\blue{Integral\:of\:\dfrac{\sqrt{tanx}}{sinx.cosx}}$

$\huge{\orange{\boxed{\fcolorbox{lime}{indigo}{\color{aqua}ANSWER}}}}$

$:\implies\:\:\Large\bf{\int\:\dfrac{\sqrt{tanx}}{sinx.cosx}\:dx}$

$:\implies\:\:\bf{\int\:\dfrac{\sqrt{tanx}}{(\frac{sinx.cosx}{cos^2x})\:cos^2x}\:dx}$

$:\implies\:\:\bf{\int\:\dfrac{\sqrt{tanx}}{(\frac{sinx}{cosx})\:cos^2x}\:dx}$

$:\implies\:\:\bf{\int\:\dfrac{\sqrt{tanx}}{tanx}\:.\:sec^2x\:dx}$

$:\implies\:\:\bf{\int\:\dfrac{1}{\sqrt{tanx}}\:.\:sec^2x\:dx}$

$\Large\bf\pink{Let}$

• $\bf{tanx\:=\:t}$

$\longmapsto\:\:\bf{sec^2x\:.\:dx\:=\:dt\:}$

$:\implies\:\:\bf{\int\:\dfrac{1}{t^{1/2}}\:.\:dt}$

$:\implies\:\:\bf{\int\:t^{-\:\frac{1}{2}}\:.\:dt}$

$:\implies\:\:\bf{\dfrac{t^{-\:\frac{1}{2}\:+\:1}}{-\:\frac{1}{2}\:+\:1}\:+\:c}$

$:\implies\:\:\bf{\dfrac{t^{\frac{1}{2}}}{\frac{1}{2}}\:+\:c}$

$:\implies\:\:\bf{2\:t^{\frac{1}{2}}\:+\:c}$

$:\implies\:\:\bf\green{2\:\sqrt{tanx}\:+\:c}$