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Question Number 167223 by cortano1 last updated on 10/Mar/22

$$\int \sin {}^3\left(3\mathrm{x}\right)\cos {}^4\left(5\mathrm{x}\right)\mathrm{dx}=?$$

Commented by greogoury55 last updated on 10/Mar/22

$$\sin {}^3\left(3x\right)=\frac{3\sin \left(3x\right)-\sin \left(9x\right)}{4}$$$$\cos {}^4\left(5x\right)=\frac{3+4\cos \left(10x\right)+\cos \left(20x\right)}{8}$$$$let\sin \left(nx\right)=s_n\mathrm{\&}\cos \left(mx\right)=c_m$$$$then{s}_n\times c_m=\frac{s_{n+m}+s_{n-m}}{2}$$

Commented by MJS_new last updated on 10/Mar/22

$${\sin }^{3}3x{\cos }^{4}5x=$$$$[\sin x=s\wedge \cos x=c]$$$$=-c^4(c^2-1){(4c^2-1)}^3{(16c^4-20c^2+5)}^{4}s$$$$\mathrm{we}\mathrm{can}\mathrm{use}t=\cos x\mathrm{and}\mathrm{get}$$$$\int t^4(t^2-1){(4t^2-1)}^3{(16t^4-20t^2+5)}^{4}dt$$$$\mathrm{which}\mathrm{has}\mathrm{to}\mathrm{be}\mathrm{expanded}...$$

Commented by cortano1 last updated on 10/Mar/22

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