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Question Number 34827 by Cheyboy last updated on 11/May/18

$$\mathit{F}\mathit{i}\mathit{n}\mathit{d}\int {\mathit{S}\mathit{i}\mathit{n}}^6\mathit{x}\mathit{d}\mathit{x}$$$$$$

Commented by rahul 19 last updated on 11/May/18

$$sirhow\sin {}^{6}xdx={\left(\frac{e^{ix}-e^{-ix}}{2i}\right)}^6?$$

Commented by Cheyboy last updated on 11/May/18

$$Siridntunderstandurmethod$$$$plzbesimplistic$$

Commented by abdo mathsup 649 cc last updated on 11/May/18

$$ihavegiventhekeybecsuse$$$$\int e^{i(2k-6)x}dx=\frac{1}{2k-6}{e}^{i(2k-6)x}+\lambda$$$$=\frac{1}{2k-6}(cos(3k-6)x+isin(3k-6)x)+\lambda .....$$

Commented by math khazana by abdo last updated on 11/May/18

$$anothermethod\left(linearization\right)$$$$\int {sin}^{6}xdx=\int {\left(\frac{e^{ix}-e^{-ix}}{2i}\right)}^{6}dx$$$$=\frac{1}{{\left(2i\right)}^6}\int \sum _{k=0}^6{C}_6^k{e}^{ikx}{e}^{-i(6-k)x}dx$$$$=\frac{1}{{\left(2i\right)}^6}\sum _{k=0}^6{C}_6^k\int e^{i(2k-6)x}dx....$$

Commented by abdo mathsup 649 cc last updated on 11/May/18

$$itsaformula\frac{e^{ix}-e^{-ix}}{2i}=\frac{2iIm\left(e^{ix}\right)}{2i}$$$$=sinx.$$

Answered by tanmay.chaudhury50@gmail.com last updated on 11/May/18

$$=\int \frac{{(1-cos2x)}^3}{2^3}$$$$\frac{1}{8}\int \frac{1-3cos2x+3{cos}^{2}2x-{cos}^{3}2x}{}$$$$\frac{1}{8}\int dx-\frac{3}{8}\int cos2xdx+\frac{3}{8}\int \frac{(1+cos4x)}{2}-\frac{1}{8}\int {cos}^{3}2x$$$$=\frac{x}{8}-\frac{3}{16}sin2x+\frac{3}{16}(x+\frac{sin4x}{4})-\frac{1}{8}\int \frac{cos6x+3cos2x}{4}$$$$=do-\frac{1}{32}\int cos6x+3cos2x$$$$=do-\frac{1}{32}\times \frac{sin6x}{6}-\frac{3}{32}\times \frac{sin2x}{2}$$$$=\frac{x}{8}-\frac{3}{16}sin2x+\frac{3}{16}(x+\frac{sin4x}{4})-\frac{1}{32}\frac{sin6x}{6}-\frac{3}{64}\times$$$$sin2x$$$$x(\frac{1}{8}+\frac{3}{16})-sin2x(\frac{3}{16}+\frac{3}{64})+sin4x\left(\frac{3}{64}\right)$$$$-\sin 6\mathrm{x}\left(\frac{1}{192}\right)$$$$=\mathrm{x}\left(\frac{5}{16}\right)-\sin 2\mathrm{x}\left(\frac{15}{64}\right)+\sin 4\mathrm{x}\left(\frac{3}{64}\right)-\frac{1}{192}\sin 6\mathrm{x}$$$$=\frac{5}{16}\mathrm{x}-\frac{15}{64}\sin 2\mathrm{x}+\frac{3}{64}\sin 4\mathrm{x}-\frac{1}{192}\sin 6\mathrm{x}$$$$$$$$$$

Commented by Cheyboy last updated on 11/May/18

$$completeitplzz$$

Commented by tanmay.chaudhury50@gmail.com last updated on 11/May/18

$$plsgothrouvhit$$

Commented by Cheyboy last updated on 11/May/18

$$\mathcal{T}hankyou$$$$$$$$$$

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