Question

integration sinx.cosx

Answer 1

first multiply and divide by 2 OK then result will be sin2theta /2 Now take 1/2 common and integrate sin2theta which is -cos2theta/2 so your final answer will be -cos2theta/4.

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Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\large\boxed{\sf{\int \:sinxcosx\:dx}}$

♣ ᴀɴꜱᴡᴇʀ :

$\boxed{\sf{\int \sin \left(x\right)\cos \left(x\right)dx=\dfrac{1}{2}\sin ^2\left(x\right)+C}}$

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

$\text { Apply u - substitution: } u=\sin (x)$

$=\int \:udu$

$\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\dfrac{x^{a+1}}{a+1},\:\quad \:a\ne -1$

$=\dfrac{u^{1+1}}{1+1}$

$\mathrm{Substitute\:back}\:u=\sin \left(x\right)$

$=\dfrac{\sin ^{1+1}\left(x\right)}{1+1}$

$\text { Simplify } \dfrac{\sin ^{1+1}(x)}{1+1}: \dfrac{1}{2} \sin ^{2}(x)$

$=\dfrac{1}{2}\sin ^2\left(x\right)$

$\mathrm{Add\:a\:constant\:to\:the\:solution}$

$\large\boxed{\sf{=\dfrac{1}{2}\sin ^2\left(x\right)+C}}$