Question

Integrate__
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Answer 1

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$\bf\Huge\red{\mid{\overline{\underline{ ANSWER }}}\mid }$

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$\Large\fbox{\color{purple}{QUESTION}}$

$\huge \int\limits \frac{1}{ \sin(x) + \sec(x) } dx$

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$\Large\fbox{\color{purple}{ SOLUTION }}$

$\pink{\implies} \int\limits \frac{1}{ \sin(x) + \sec(x) } dx \\ \\ \pink{\implies} \int\limits \frac{1}{ \sin(x) + \frac{1}{ \cos(x) } } dx \\ \\ \pink{\implies} \int\limits \frac{ \cos(x) }{1 + \sin(x) \cos(x) } dx$

➠ ᴍᴜʟᴛɪᴘʟʏɪɴɢ ʙᴏᴛʜ ɴᴜᴍ. ᴀɴᴅ ᴅᴇɴ. ʙʏ 2 ᴡᴇ ɢᴇᴛ,

$\pink{\implies} \int\limits \frac{2 \cos(x) }{2 \sin(x) \cos(x) + 2} \\ \\ \pink{\implies} \int\limits \frac{( \cos(x) - \sin(x) ) + ( \cos(x) + \sin(x) ) }{1 + 1 + 2 \sin(x) \cos(x) } dx \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{{ \sin(x) }^{2} + { \cos(x) }^{2} = 1 } \\ \\ \pink{\implies} \int\limits \frac{( \cos(x) - \sin(x)) + ( \cos(x) + \sin(x) ) }{1 + {( \sin(x) + \cos(x) ) }^{2} } \\ \\ \pink{\implies} \int\limits \frac{( \cos(x) - \sin(x)) }{1 + {( \sin(x) + \cos(x)) }^{2} } dx + \int\limits \frac{( \cos(x) + \sin(x)) }{1 + {( \sin(x) + \cos(x) )}^{2} } dx \\ \\ \pink{\implies} \int\limits \frac{( \cos(x) - \sin(x)) }{1 + {( \sin(x) + \cos(x) }^{2} } dx + \int\limits \frac{( \cos(x) + \sin(x) )}{1 + {2 - ( { \sin(x) - \cos(x)) }^{2}} } dx \\ \\ \pink{\implies} \int\limits \underbrace{\bf\red{{\bigg[ \frac{( \cos(x) - \sin(x) )}{1 + {( \sin(x) + \cos(x) )}^{2} } \bigg]}}}dx + \int\limits \underbrace{\bf\red{{\bigg[ \frac{( \cos(x) + \sin(x)) }{3 - ( {\sin(x) - \cos(x)}^{2} ) } \bigg]}}}dx \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: I_1 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: I_2$

➠ ʟᴇᴛ ɪɴ ɪ1

➻$\sin(x) + \cos(x) = t \\ {\cos(x) - \sin(x) }dx = dt$

➠ ʟᴇᴛ ɪɴ ɪ2

➻ $\sin(x) - \cos(x) = v \\ { \cos(x) + \sin(x) }dx = dv$

➠ ᴏɴ sᴜʙsᴛɪᴛᴜᴛɪɴɢ, .

$\pink{\implies} \int\limits \frac{dt}{1 + {t}^{2} } + \int\limits \frac{dv}{3 - {v}^{2} } \\ \\ \pink{\implies} { \tan }^{-1} t \: + \frac{1}{2 \sqrt{3} } log \bigg| \frac{ \sqrt{3} - v }{ \sqrt{3} + v} \bigg| + C \\ \\ \green{\implies} \pink{ { \tan}^{-1}( \sin(x) + \cos(x) ) } \pink{+ \frac{1}{2 \sqrt{3} }} \pink{ log \bigg| \frac{ \sqrt{3} - \sin(x) + \cos(x) }{ \sqrt{3} + \sin(x) - \cos(x) } \bigg| + C}$

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$\bf\Large\red{ THANKS \: FOR \: YOUR}$

$\bf\Large\red{ QUESTION \: HOPE \: IT }$

$\bf\Large\red{ HELPS }$

$\Large\mathcal\green{FOLLOW \: ME}$

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Answer 2

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$\huge\boxed{\underline{\mathcal{\red{G} \green{i} \pink{v} \orange{e} \blue{n}}}}$

$</p><p>\huge \int\limits \frac{1}{ \sin(x) + \sec(x) } dx∫ </p><p>sin(x)+sec(x)</p><p>1</p><p> </p><p> dx$

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$\bf\large\underline\red{SoLUtiON}$

$$\begin{lgathered}\blue{\implies} \int\limits \frac{1}{ \sin(x) + \sec(x) } dx \\ \\ \blue{\implies} \int\limits \frac{1}{ \sin(x) + \frac{1}{ \cos(x) } } dx \\ \\ \blue{\implies} \int\limits \frac{ \cos(x) }{1 + \sin(x) \cos(x) } dx\end{lgathered}$$

ᴍᴜʟᴛɪᴘʟʏɪɴɢ ʙᴏᴛʜ ɴᴜᴍ. ᴀɴᴅ ᴅᴇɴ. ʙʏ 2 ᴡᴇ ɢᴇᴛ,

$$\begin{lgathered}\blue{\implies} \int\limits \frac{2 \cos(x) }{2 \sin(x) \cos(x) + 2} \\ \\ \blue{\implies} \int\limits \frac{( \cos(x) - \sin(x) ) + ( \cos(x) + \sin(x) ) }{1 + 1 + 2 \sin(x) \cos(x) } dx \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{{ \sin(x) }^{2} + { \cos(x) }^{2} = 1 } \\ \\ \blue{\implies} \int\limits \frac{( \cos(x) - \sin(x)) + ( \cos(x) + \sin(x) ) }{1 + {( \sin(x) + \cos(x) ) }^{2} } \\ \\ \blue{\implies} \int\limits \frac{( \cos(x) - \sin(x)) }{1 + {( \sin(x) + \cos(x)) }^{2} } dx + \int\limits \frac{( \cos(x) + \sin(x)) }{1 + {( \sin(x) + \cos(x) )}^{2} } dx \\ \\ \blue{\implies} \int\limits \frac{( \cos(x) - \sin(x)) }{1 + {( \sin(x) + \cos(x) }^{2} } dx + \int\limits \frac{( \cos(x) + \sin(x) )}{1 + {2 - ( { \sin(x) - \cos(x)) }^{2}} } dx \\ \\ \blue{\implies} \int\limits \underbrace{\bf\purple{{\bigg[ \frac{( \cos(x) - \sin(x) )}{1 + {( \sin(x) + \cos(x) )}^{2} } \bigg]}}}dx + \int\limits \underbrace{\bf\purple{{\bigg[ \frac{( \cos(x) + \sin(x)) }{3 - ( {\sin(x) - \cos(x)}^{2} ) } \bigg]}}}dx \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: I_1 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: I_2\end{lgathered}$$

➨ ʟᴇᴛ ɪɴ ɪ1

❧ $$\begin{lgathered}\sin(x) + \cos(x) = t \\ {\cos(x) - \sin(x) }dx = dt\end{lgathered}$$

➨ʟᴇᴛ ɪɴ ɪ2

❧ $$\begin{lgathered}\sin(x) - \cos(x) = v \\ { \cos(x) + \sin(x) }dx = dv\end{lgathered}$$[/tex]

➨ ᴏɴ sᴜʙsᴛɪᴛᴜᴛɪɴɢ, .

$$\begin{lgathered}\blue{\implies} \int\limits \frac{dt}{1 + {t}^{2} } + \int\limits \frac{dv}{3 - {v}^{2} } \\ \\ \blue{\implies} { \tan }^{-1} t \: + \frac{1}{2 \sqrt{3} } log \bigg| \frac{ \sqrt{3} - v }{ \sqrt{3} + v} \bigg| + C \\ \\ \blue{\implies} \purple{ { \tan}^{-1}( \sin(x) + \cos(x) ) } \purple{+ \frac{1}{2 \sqrt{3} }} \purple{ log \bigg| \frac{ \sqrt{3} - \sin(x) + \cos(x) }{ \sqrt{3} + \sin(x) - \cos(x) } \bigg| + C}\end{lgathered}$$

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$</p><p>\huge{\orange{\boxed{\boxed{\pink{\underline{\green{\mathscr{Hope\ it\ helps\ u}}}}}}}}$

$\huge \underline \bold \red {Follow \: me}$

$<marquee>Mark it as brainliest</marquee>$

$\red {Be \: Brainly}$

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$<svg width="250" height="250" viewBox="0 0 100 100">\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ <path fill="pink" d="M92.71,7.27L92.71,7.27c-9.71-9.69-25.46-9.69-35.18,0L50,14.79l-7.54-7.52C32.75-2.42,17-2.42,7.29,7.27v0 c-9.71,9.69-9.71,25.41,0,35.1L50,85l42.71-42.63C102.43,32.68,102.43,16.96,92.71,7.27z"></path>\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ <animateTransform \ \textless \ br /\ \textgreater \ attributeName="transform" \ \textless \ br /\ \textgreater \ type="scale" \ \textless \ br /\ \textgreater \ values="1; 1.5; 1.25; 1.5; 1.5; 1;" \ \textless \ br /\ \textgreater \ dur="2s" \ \textless \ br /\ \textgreater \ repeatCount="40"> \ \textless \ br /\ \textgreater \ </animateTransform>\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ </svg>$