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integrating: 

\(\int dx ~ 2x \sin^{3}(x) \)

 

1. Let

\(\begin{array}{|rcll|} \hline \int dx ~ \sin^3(x) &=& \int dx ~ \sin^2(x)\cdot \sin(x) \\ &=& \int dx ~ [ 1-\cos^2(x) ]\cdot \sin(x) \\ && u = \cos(x) \\ && du = -\sin(x) dx \\ &=& \int du ~ \frac{1}{-\sin(x)} \cdot (1-u^2) \cdot \sin(x) \\ &=& -\int du ~ (1-u^2) \\ &=& -\int du +\int du ~ u^2 \\ &=& -u + \frac13 \cdot u^3 \\ &=& \mathbf{-\cos(x) + \frac13 \cdot \cos^3(x)} \\ \hline \end{array} \)

 

2. Let

\(\begin{array}{|rcll|} \hline \int dx ~ \cos^3(x) &=& \int dx ~ \cos^2(x)\cdot \cos(x) \\ &=& \int dx ~ [ 1-\sin^2(x) ]\cdot \cos(x) \\ && u = \sin(x) \\ && du = \cos(x) dx \\ &=& \int du ~ \frac{1}{\cos(x)} \cdot (1-u^2) \cdot \cos(x) \\ &=& \int du ~ (1-u^2) \\ &=& \int du - \int du ~ u^2 \\ &=& u - \frac13 \cdot u^3 \\ &=& \mathbf{\sin(x) - \frac13 \cdot \sin^3(x)} \\ \hline \end{array}\)

 

3. Let

\(\begin{array}{|rcll|} \hline \int dx ~ x \cdot \sin^3(x) \\ && u = x \\ && u' = 1 \\ && v = -\cos(x) + \frac13 \cdot \cos^3(x) \\ && v' = \sin^3(x) \\ &=& x \cdot [-\cos(x) + \frac13 \cdot \cos^3(x)] - \int dx ~ 1\cdot [-\cos(x) + \frac13 \cdot \cos^3(x)] \\ &=& -x \cdot \cos(x) + \frac13 \cdot x \cdot \cos^3(x) + \int dx ~ \cos(x) - \int dx ~ \frac13 \cdot \cos^3(x) \\ &=& -x \cdot \cos(x) + \frac13 \cdot x \cdot \cos^3(x) + \sin(x) - \frac13 \cdot \int dx ~ \cos^3(x) \\ &=& -x \cdot \cos(x) + \frac13 \cdot x \cdot \cos^3(x) + \sin(x) - \frac13 \cdot [ \sin(x) - \frac13 \cdot \sin^3(x)] \\ &=& -x \cdot \cos(x) + \frac13 \cdot x \cdot \cos^3(x) + \sin(x) - \frac13 \cdot \sin(x) + \frac19 \cdot \sin^3(x)] \\ &=& -x \cdot \cos(x) + \frac13 \cdot x \cdot \cos^3(x) + \frac23 \cdot \sin(x) + \frac19 \cdot \sin^3(x) \\\\ \int dx ~ 2\cdot x \cdot \sin^3(x) &=& -2x \cdot \cos(x) + \frac23 \cdot x \cdot \cos^3(x) + \frac43 \cdot \sin(x) + \frac29 \cdot \sin^3(x) \\ &=&\mathbf{\frac{1}{9} \cdot [ -18x \cdot \cos(x) + 6 x \cdot \cos^3(x) + 12 \cdot \sin(x) + 2 \cdot \sin^3(x) ] }\\ \hline \end{array} \)

 

4. Let

\(\begin{array}{|rcll|} \hline && \cos^3(x) = \frac14 \cdot [3\cdot \cos(x) + \cos(3x) ] \\ && \sin^3(x) = \frac14 \cdot [3\cdot \sin(x) - \sin(3x) ] \\ \hline \end{array}\)

 

5. Let

\(\begin{array}{|rcll|} \hline && \int dx ~ 2\cdot x \cdot \sin^3(x) \\ &=& \frac{1}{9} \cdot [ -18x \cdot \cos(x) + 6 x \cdot \cos^3(x) + 12 \cdot \sin(x) + 2 \cdot \sin^3(x) ] \\ &=& \frac{1}{9} \cdot \{ -18x \cdot \cos(x) + 6 x \cdot \frac14 \cdot [3\cdot \cos(x) + \cos(3x) ] + 12 \cdot \sin(x) + 2 \cdot \frac14 \cdot [3\cdot \sin(x) - \sin(3x) ] \} \\ &=& -2x \cdot \cos(x) + \frac16 \cdot x\cdot [3\cdot \cos(x) + \cos(3x) ] + \frac{4}{3} \cdot \sin(x) + \frac{1}{18} \cdot [3\cdot \sin(x) - \sin(3x) ] \\ &=& -2x \cdot \cos(x) + \frac12 \cdot x\cdot \cos(x) + \frac16 \cdot x\cdot \cos(3x) + \frac{4}{3} \cdot \sin(x) + \frac{1}{6} \cdot \sin(x) - \frac{1}{18} \cdot \sin(3x) \\ &=& \mathbf{ - \frac32 \cdot x\cdot \cos(x) + \frac16 \cdot x\cdot \cos(3x) + \frac32 \cdot \sin(x) - \frac{1}{18} \cdot \sin(3x) }\\ \hline \end{array} \)

 

or

\(\begin{array}{|rcll|} \hline && \int dx ~ 2\cdot x \cdot \sin^3(x) \\ &=& - \frac32 \cdot x\cdot \cos(x) + \frac16 \cdot x\cdot \cos(3x) + \frac32 \cdot \sin(x) - \frac{1}{18} \cdot \sin(3x) \\ &=& \mathbf{ \frac{1}{18} [ - 27 \cdot x\cdot \cos(x) + 3 \cdot x\cdot \cos(3x) + 27 \cdot \sin(x) - \sin(3x) ] } \\ \hline \end{array}\)

 

laugh

Oct 5, 2016

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