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Next Problem ssignment 3 ubstitution by parts: blem 10 t) ate the indefinite integral. xsin^(2)(5x)dx=◻+C. Hint: Integrate by parts with u=x.

Next Problem\newlinessignment 33 ubstitution by parts: blem 1010\newlinet)\newlineate the indefinite integral.\newlinexsin⁡2(5x)dx=□+C. x \sin ^{2}(5 x) d x=\square+C . \newlineHint: Integrate by parts with u=x u=x .

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Q. Next Problem\newlinessignment 33 ubstitution by parts: blem 1010\newlinet)\newlineate the indefinite integral.\newlinexsin⁡2(5x)dx=□+C. x \sin ^{2}(5 x) d x=\square+C . \newlineHint: Integrate by parts with u=x u=x .
  1. Choose uu and dvdv: Choose uu and dvdv for integration by parts.\newlineLet u=xu = x, then du=dxdu = dx.\newlineLet dv=sin⁡2(5x)dxdv = \sin^{2}(5x)dx, then we need to find vv.
  2. Find vv using trigonometric identity: To find vv, integrate dvdv. Integrating sin⁡2(5x)\sin^{2}(5x) requires using a trigonometric identity. sin⁡2(5x)=1−cos⁡(10x)2\sin^{2}(5x) = \frac{1 - \cos(10x)}{2}. Now integrate 1−cos⁡(10x)2dx\frac{1 - \cos(10x)}{2} dx to find vv.
  3. Integrate (1−cos⁡(10x))/2 dx(1 - \cos(10x))/2 \, dx: Integrate (1−cos⁡(10x))/2 dx(1 - \cos(10x))/2 \, dx.
    v=∫(12−cos⁡(10x)2) dxv = \int(\frac{1}{2} - \frac{\cos(10x)}{2}) \, dx
    v=(12)x−(120)sin⁡(10x)+Cv = (\frac{1}{2})x - (\frac{1}{20})\sin(10x) + C.
  4. Apply integration by parts formula: Apply integration by parts formula: ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du.∫xsin⁡2(5x) dx=uv−∫v du\int x\sin^{2}(5x)\,dx = uv - \int v\,du= x\left(\frac{\(1\)}{\(2\)}x - \frac{\(1\)}{\(20\)}\sin(\(10x)\right) - \int\left(\frac{11}{22}x - \frac{11}{2020}\sin(1010x)\right) dx

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