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Evaluate the integral\newline ∫cos⁡3(x)sin⁡(x)dx=\int \cos^{3}(x)\sin(x)dx =

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Full solution

Q. Evaluate the integral\newline ∫cos⁡3(x)sin⁡(x)dx=\int \cos^{3}(x)\sin(x)dx =
  1. Rewrite integral using substitution: Rewrite the integral using a substitution method. Let u=cos⁡(x)u = \cos(x), then du=−sin⁡(x)dxdu = -\sin(x)dx.
  2. Substitute uu and dudu: Substitute uu and dudu into the integral: ∫u3(−du)=−∫u3 du\int u^3(-du) = -\int u^3\,du.
  3. Integrate u3u^3: Integrate u3u^3 with respect to uu: −∫u3 du=−u44+C-\int u^3 \, du = -\frac{u^4}{4} + C.
  4. Substitute back for uu: Substitute back cos⁡(x)\cos(x) for uu: −cos⁡4(x)4+C-\frac{\cos^4(x)}{4} + C.

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