Question

${\displaystyle \int(( \sin \: x +a \sec \: x)) {}^{2} }dx$


❖ᴏɴʟʏ ᴘʀᴏᴘᴇʀ ꜱᴏʟᴠᴇᴅ ᴀɴꜱᴡᴇʀ ᴡɪᴛʜ ɢᴏᴏᴅ ᴇxᴘʟᴀɴᴀɪᴏɴ ɴᴇᴇᴅᴇᴅ
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Answer 1

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∫((sinx+asecx))2dx

$\huge\color{blue}\boxed{\colorbox{lightgreen}{♧Ânswer♧}}\bigstar$

We should try to use substitution by setting u=cosx, so du=−sinxdx.

This gives us the integral:

∫sinxcos2xdx=−∫−sinxcos2xdx=−∫1u2du=−∫u−2du

From here, use the rule

∫undu=un+1n+1+C

Thus,

−∫u−2du=−u−1−1+C=1u+C

=1cosx+C=secx+C

Not know I am best user or not but anyway.

$⚘Thankyou$

Answer 2

Identities Used :-

$\boxed{ \tt{ \: \displaystyle \int {sec}^{2}x \: dx = tanx + c \: }}$

$\boxed{ \tt{ \: \displaystyle \int tanxdx = - \: log |cosx| + c \: }}$

$\boxed{ \tt{ \: \displaystyle \int \: dx \: = x + c \: }}$

$\boxed{ \tt{ \: \displaystyle \int \: {x}^{n} \: dx \: = \frac{ {x}^{n + 1} }{n + 1} + c \: }}$

$\boxed{ \tt{ \: \displaystyle \int \: sinx \: dx \: = \: - \: cosx + c \: }}$

Let's solve the problem now!!

$\green{\large\underline{\sf{Solution-}}}$

Given integral is

$\rm :\longmapsto\:\displaystyle \int \: {(sinx + asecx)}^{2} \: dx$

We know,

$\boxed{ \tt{ \: {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} \: }}$

So, using this

$\rm \:  =  \: \displaystyle \int \bigg[ {sin}^{2}x + {a}^{2} {sec}^{2}x + 2asinxsecx \bigg] \: dx$

$\rm \:  =  \: \displaystyle \int {sin}^{2}xdx + {a}^{2} \displaystyle \int {sec}^{2}x \: dx + 2x \displaystyle \int \frac{sinx}{cosx} \: dx$

$\rm \:  =  \displaystyle \int \frac{1 - cos2x}{2}dx + {a}^{2}tanx + 2a \displaystyle \int tanx \: dx$

$\rm \:  =  \:\dfrac{1}{2} \displaystyle \int [1 - cos2x]dx + {a}^{2}tanx - log |cosx| + c$

$\red{\rm \:  =  \:\dfrac{1}{2} \bigg[x - \dfrac{sin2x}{2} \bigg] + {a}^{2}tanx - log |cosx| + c}$

Explore more :-

$\blue{\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}}$