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Use a suitable change of variables to determine the indefinite integral. (Use C for the constant of integration.) int(sin^(2)(theta)-2sin(theta))(sin^(3)(theta)-3sin^(2)(theta))^(3)cos(theta)d theta

Use a suitable change of variables to determine the indefinite integral. (Use C C for the constant of integration.)\newline∫(sin⁡2(θ)−2sin⁡(θ))(sin⁡3(θ)−3sin⁡2(θ))3cos⁡(θ)dθ \int\left(\sin ^{2}(\theta)-2 \sin (\theta)\right)\left(\sin ^{3}(\theta)-3 \sin ^{2}(\theta)\right)^{3} \cos (\theta) d \theta

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Q. Use a suitable change of variables to determine the indefinite integral. (Use C C for the constant of integration.)\newline∫(sin⁡2(θ)−2sin⁡(θ))(sin⁡3(θ)−3sin⁡2(θ))3cos⁡(θ)dθ \int\left(\sin ^{2}(\theta)-2 \sin (\theta)\right)\left(\sin ^{3}(\theta)-3 \sin ^{2}(\theta)\right)^{3} \cos (\theta) d \theta
  1. Define uu and dudu: Let u=sin⁡3(θ)−3sin⁡2(θ)u = \sin^3(\theta) - 3\sin^2(\theta), then du=(3sin⁡2(θ)cos⁡(θ)−6sin⁡(θ)cos⁡(θ))dθdu = (3\sin^2(\theta)\cos(\theta) - 6\sin(\theta)\cos(\theta))d\theta.
  2. Rewrite in terms of uu: Rewrite the integral in terms of uu: ∫((sin⁡2(θ)−2sin⁡(θ))⋅u3⋅cos⁡(θ)) dθ=∫((sin⁡2(θ)−2sin⁡(θ))⋅(sin⁡3(θ)−3sin⁡2(θ))3⋅cos⁡(θ)) dθ\int((\sin^2(\theta) - 2\sin(\theta)) \cdot u^3 \cdot \cos(\theta))\,d\theta = \int((\sin^2(\theta) - 2\sin(\theta)) \cdot (\sin^3(\theta) - 3\sin^2(\theta))^3 \cdot \cos(\theta))\,d\theta.
  3. Substitute dudu and uu: Substitute dudu and uu into the integral: ∫((sin⁡2(θ)−2sin⁡(θ))⋅u3⋅cos⁡(θ))dθ=∫(u3⋅(3sin⁡2(θ)−6sin⁡(θ)))dθ\int((\sin^2(\theta) - 2\sin(\theta)) \cdot u^3 \cdot \cos(\theta))d \theta = \int(u^3 \cdot (3\sin^2(\theta) - 6\sin(\theta)))d \theta.
  4. Factor out constants: Factor out constants from dudu: du=3(sin⁡2(θ)−2sin⁡(θ))cos⁡(θ)dθdu = 3(\sin^2(\theta) - 2\sin(\theta))\cos(\theta) d\theta.
  5. Integrate u3u^3: Now the integral becomes ∫(u3⋅(13) du)=(13)∫(u3 du)\int(u^3 \cdot (\frac{1}{3})\,du) = (\frac{1}{3})\int(u^3 \,du).
  6. Substitute back for u: Integrate u3u^3 with respect to uu: (1/3)∫(u3 du)=(1/3)(u4/4)+C(1/3)\int(u^3 \, du) = (1/3)(u^4/4) + C.
  7. Simplify the expression: Substitute back for uu: 13(u44)+C=13((sin⁡3(θ)−3sin⁡2(θ))44)+C\frac{1}{3}\left(\frac{u^4}{4}\right) + C = \frac{1}{3}\left(\frac{(\sin^3(\theta) - 3\sin^2(\theta))^4}{4}\right) + C.
  8. Simplify the expression: Substitute back for u: (13)(u44)+C=(13)((sin⁡3(θ)−3sin⁡2(θ))4/4)+C.(\frac{1}{3})(\frac{u^4}{4}) + C = (\frac{1}{3})((\sin^3(\theta) - 3\sin^2(\theta))^4/4) + C. Simplify the expression: (13)((sin⁡3(θ)−3sin⁡2(θ))4/4)+C=(112)(sin⁡3(θ)−3sin⁡2(θ))4+C.(\frac{1}{3})((\sin^3(\theta) - 3\sin^2(\theta))^4/4) + C = (\frac{1}{12})(\sin^3(\theta) - 3\sin^2(\theta))^4 + C.

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