Question

integration [ (cosx+sinx) (1-1/2 sin2x) sec^2 x cosec^2 x dx]​

Answer 1

$\huge\sf\underline{\red{Solution\;:}}$

$\longmapsto\displaystyle\int \sf\green{ (\cos x + \sin x )(1-\frac{1}{2}\sin2x )(\sec^2x cosec^2x )dx}\\\\\implies\sf\displaystyle\int \sf (\cos x + \sin x )(1-\frac{1}{\cancel{2}}\times\cancel{2}\sin x\cos x )(\sec^2xcosec^2x )dx\\\\\implies\displaystyle\int \sf (\cos x + \sin x )(1-\sin x\cos x )(\sec^2xcosec^2x )dx\\\\\implies\displaystyle\int \sf (\cos x + \sin x )(\sin^2x+\cos^2x-\sin x\cos x )(\sec^2xcosec^2x )dx\\\\\implies\sf\displaystyle\int \sf ( \sin^3 x + \cos^3 x)(\sec^2x\;cosec^2x )dx\\\\\implies\sf\displaystyle\int \sf ( \sin^3 x + \cos^3 x)(\frac{1}{\cos^2x}\frac{1}{\sin^2x})dx\\\\\implies\sf\displaystyle\int \sf ( \sin^3 x )(\frac{1}{\cos^2x}\frac{1}{\sin^2x})+ (\cos^3 x)(\frac{1}{\cos^2x}\frac{1}{\sin^2x})dx\\\\\\\implies\sf\displaystyle\int \sf(\frac{\sin x}{\cos^2x})+(\frac{\cos x}{\sin^2x})dx$

$\textbf{We can solve from here in 2 ways.}$

$\sf \purple{\underbrace{\sf Method}\: 1}:$

$\\\\\implies\sf\displaystyle\int \sf(\frac{\sin x}{\cos x}).\frac{1}{\cos x}+(\frac{\cos x}{\sin x}).\frac{1}{\sin x}dx$

$\\\\\implies\sf\displaystyle\int \sf\tan x.\sec x + \cot x.cosec x.dx\\\\\implies\sf\displaystyle\int \sf\tan x.\sec x + \displaystyle\int\cot x.cosec x.dx$

so,

$\implies \sf \sec x - cosec x + c$

$\sf \purple{\underbrace{\sf Method} \:2}:$

$\\\\\implies\sf\displaystyle\int \sf(\frac{\sin x}{\cos^2x})+\displaystyle\int(\frac{\cos x}{\sin^2x})dx\\\\\implies\sf\displaystyle\int \sf(\frac{-d(\cos x)}{\cos^2x})+\displaystyle\int(\frac{d(\sin x)}{\sin^2x})dx\\\\\implies\sf\displaystyle-\int {\cos x}^{-2}.d(\cos x)+\displaystyle\int{\sin^x}^{-2}.d(\sin x)dx\\\\\implies\sf\displaystyle-\dfrac{{\cos x}^{-2+1}}{-2+1}+\dfrac{{\sin x}^{-2+1}}{-2+1}+c\\\\\implies \sf\displaystyle-\dfrac{{\cos x}^{-1}}{-1}+\dfrac{{\cos x}^{-1}}{-1}+c\\\\\implies \displaystyle \sf\sec x - cosec x+c\\\\\implies\sf\displaystyle \sf\purple{ - cosec x +\sec x+ c}$

$\bf\large \underline{Therefore},$

$\displaystyle\int \sf (\cos x + \sin x )(\frac{1}{2}\sin2x )(\sec^2xcosec^2x )dx =- cosec x + \sec x + c$

$\:\:\:\:\:\:\green{\textsf{Option c is correct answer}}$

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$\\\large\red{\sf\underbrace{\sf Formulas\: used}}:$

$\red\bigstar\;\;\boxed{\green{\sin2x= 2\sin x\cos x}}$

$\\\\\red\bigstar\;\;\boxed{\green{\sin^2\theta+\cos^2\theta = 1}}$

$\\\red\bigstar\;\;\boxed{\sf\green{(a+b)(a^2+b^2-ab)=a^3+=b^3}}$

$\\\red\bigstar\;\;\boxed{\green{\sec\theta=\frac{1}{\cos\theta}}}\;\;\;\;\;\;\;\;\;\;\;\\\red\bigstar\;\;\boxed{\green{cosec\theta=\frac{1}{\sin\theta}}}$

$\\\bigstar\;\;\boxed{\sec\theta=\frac{1}{\cos\theta}}\;\;\;\;\;\;\;\;\;\;\;\bigstar\;\;\boxed{cosec\theta=\frac{1}{\sin\theta}}$

$\\\\\bigstar\;\;\boxed{\tan\theta=\frac{\sin\theta}{\cos\theta}}\;\;\;\;\;\;\;\;\;\;\;\bigstar\;\;\boxed{\cot\theta=\frac{\cos\theta}{\sin\theta}}$

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