Question

Integrate the function
$\huge\tt\frac{ \sqrt{tanx} }{sinxcosx}$​

Answer 1

No one knows everything, but everyone knows something. With Brainly, students combine their strengths and talents to tackle problems together.

$\huge{\bold☘}\mathfrak\pink{\bold{\underline{{ ℘ɧεŋσɱεŋศɭ}}}}{\bold☘}$

$\red{\bold{\underline{\underline{❥Question᎓}}}}$ Integrate the function

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

$\huge\tt{\boxed{\overbrace{\underbrace{\blue{Answer</p><p> }}}}}$

╔════════════════════════╗

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _✍️

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

$\huge\tt\bold{☛Let tanx=t}$

$\huge\tt\bold{☛differentiating \:both\: sides \:w.r.t.x}$

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

ㅤ ㅤ ㅤ ㅤ ㅤ

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

╚════════════════════════╝

нσρє ıт нєłρs yσυ

_____________________

тнαηkyσυ

Answer 2

Step-by-step explanation:

No one knows everything, but everyone knows something. With Brainly, students combine their strengths and talents to tackle problems together.

\huge{\bold☘}\mathfrak\pink{\bold{\underline{{ ℘ɧεŋσɱεŋศɭ}}}}{\bold☘}☘

℘ɧεŋσɱεŋศɭ

☘

\red{\bold{\underline{\underline{❥Question᎓}}}}

❥Question᎓

Integrate the function

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

sinxcosx

tanx

\huge\tt{\boxed{\overbrace{\underbrace{\blue{Answer }}}}}

Answer

╔════════════════════════╗

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _✍️

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

sinxcosx

tanx

ㅤ ㅤ ㅤ ㅤ ㅤ

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

sinxcosx×

cosx

cosx

tanx

ㅤ ㅤ ㅤ ㅤ ㅤ

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

sinx×

cosx

cos

2

x

tanx

ㅤ ㅤ ㅤ ㅤ ㅤ

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

cos

2

x×

cosx

sinx

tanx

ㅤ ㅤ ㅤ ㅤ ㅤ

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

cos

2

x×tanx

tanx

ㅤ ㅤ ㅤ ㅤ ㅤ

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

2

1

−1

×

cos

2

x

1

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)

−

2

1

×

cos

2

x

1

=(tanx)

−

2

1

×sec

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 \times dx⇛(tan)

−

2

1

×

cos

2

x

1

=∫(tanx)

−

2

1

×sec

2

x×dx

ㅤ ㅤ ㅤ ㅤ ㅤ

\huge\tt\bold{☛Let tanx=t}☛Lettanx=t

\huge\tt\bold{☛differentiating \:both\: sides \:w.r.t.x}☛differentiatingbothsidesw.r.t.x

ㅤ ㅤ ㅤ ㅤ ㅤ

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

2

x=

dx

dt

ㅤ ㅤ ㅤ ㅤ ㅤ

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

sec

2

x

dt

ㅤ ㅤ ㅤ ㅤ ㅤ

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

−

2

1

×sec

2

x×dx

ㅤ ㅤ ㅤ ㅤ ㅤ

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

−

2

1

×sec

2

x×

sec

2

x

dt

ㅤ ㅤ ㅤ ㅤ ㅤ

⇛ \huge\tt ∫ {t}^{ - \frac{1}{2} } \times dt⇛∫t

−

2

1

×dt

ㅤ ㅤ ㅤ ㅤ ㅤ

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

−

2

1

+1

t

−

2

1

+1

+c

ㅤ ㅤ ㅤ ㅤ ㅤ

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

2

1

t

2

1

+c=2t

2

1

+c=2

t

+c

ㅤ ㅤ ㅤ ㅤ ㅤ

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

t

+c=2

tanx

+c

╚════════════════════════╝

нσρє ıт нєłρs yσυ

_____________________

тнαηkyσυ