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blem 10 t) ate the indefinite integral. sin^(2)(5x)dx=◻+C Integrate by parts with u=x. gauss.vaniercollege.qc.ca

blem 1010\newlinet)\newlineate the indefinite integral.\newlinesin⁡2(5x)dx=□+C \sin ^{2}(5 x) d x=\square+C \newlineIntegrate by parts with u=x u=x .\newlinegauss.vaniercollege.qc.ca

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Q. blem 1010\newlinet)\newlineate the indefinite integral.\newlinesin⁡2(5x)dx=□+C \sin ^{2}(5 x) d x=\square+C \newlineIntegrate by parts with u=x u=x .\newlinegauss.vaniercollege.qc.ca
  1. Identify integral: Identify the integral to solve. ∫sin⁡2(5x) dx\int \sin^{2}(5x)\,dx
  2. Use trigonometric identity: Use the trigonometric identity sin⁡2(θ)=1−cos⁡(2θ)2\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} to simplify the integral.\newline∫1−cos⁡(10x)2 dx\int \frac{1 - \cos(10x)}{2} \, dx
  3. Split into two integrals: Split the integral into two separate integrals.\newline(\frac{\(1\)}{\(2\)})\int dx - (\frac{\(1\)}{\(2\)})\int \cos(\(10x)dx
  4. Integrate constant part: Integrate the first part, which is a constant with respect to xx.12x\frac{1}{2}x
  5. Integrate cos⁡(10x)\cos(10x): Integrate the second part using the basic integral of cos⁡(ax)\cos(ax).\newline−120sin⁡(10x)-\frac{1}{20}\sin(10x)
  6. Combine and add constant: Combine the two parts and add the constant of integration CC.12x−120sin⁡(10x)+C\frac{1}{2}x - \frac{1}{20}\sin(10x) + C

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