Question #193038

Cosydy/dxCosy dy/dx = Sin2xCosxSin^2x Cosx


1
Expert's answer
2021-05-14T11:50:15-0400

Given,

cosydydx=sin2x×cosx cosy×dy=sin2x×cosxdxcosy\cdot \dfrac{dy}{dx}=sin^2x\times cosx\\\ \\cosy\times dy=sin^2x\times cosxdx


Now, Integrating both sides

cosydy=sin2xcosxdx\int cosydy=\int sin^2xcosxdx

Let I1=sin2xcosxdxLet \ I_1=\int sin^2xcosxdx\\

Substitute sinx = t

then cosxdx = dt

I1=t2dtI1=t33I_1=\int t^2dt\\I_1=\frac{t^3}{3}\\


Now, I1=13sin3xI_1= \dfrac{1}{3}sin^3x


So,

cosydy=sin2xcosxdx\int cosydy=\int sin^2xcosxdx

siny=13sin3x+Cy=sin1(13sin3x+C)siny =\dfrac{1}{3}sin^3x +C\\\boxed{y=sin^{-1}(\frac{1}{3}sin^3x+C)}


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