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Integration Questions

Question Number 121611 by benjo_mathlover last updated on 10/Nov/20

$$\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:\mathrm{sin}\:\left(\mathrm{2}\pi\mathrm{x}\right)\mathrm{cos}\:\left(\mathrm{5}\pi\mathrm{x}\right)\:\mathrm{dx}\:?\: \\$$

Commented by mr W last updated on 10/Nov/20

$$\int_{\mathrm{0}} ^{\mathrm{2}} \mathrm{sin}\:\left(\mathrm{2}\pi{x}\right)\mathrm{cos}\:\left(\mathrm{5}\pi{x}\right){dx} \\$$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{2}} \left[\mathrm{sin}\:\left(\mathrm{7}\pi{x}\right)−\mathrm{sin}\:\left(\mathrm{3}\pi{x}\right)\right]{dx} \\$$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[−\frac{\mathrm{1}}{\mathrm{7}\pi}\mathrm{cos}\:\left(\mathrm{7}\pi{x}\right)+\frac{\mathrm{1}}{\mathrm{3}\pi}\mathrm{cos}\:\left(\mathrm{3}\pi{x}\right)\right]_{\mathrm{0}} ^{\mathrm{2}} \\$$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[−\frac{\mathrm{1}}{\mathrm{7}\pi}\left(\mathrm{1}−\mathrm{1}\right)+\frac{\mathrm{1}}{\mathrm{3}\pi}\left(\mathrm{1}−\mathrm{1}\right)\right] \\$$$$=\mathrm{0} \\$$

Commented by MJS_new last updated on 10/Nov/20

$$\mathrm{sin}\:\left(\mathrm{2}\pi\left(\mathrm{2}−{x}\right)\right)=−\mathrm{sin}\:\left(\mathrm{2}\pi{x}\right) \\$$$$\mathrm{cos}\:\left(\mathrm{5}\pi\left(\mathrm{2}−{x}\right)\right)=\mathrm{cos}\:\left(\mathrm{5}\pi{x}\right) \\$$$$\Rightarrow\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}...=\mathrm{0} \\$$

Answered by MJS_new last updated on 10/Nov/20

$$=\mathrm{0} \\$$

Commented by benjo_mathlover last updated on 10/Nov/20

$$\mathrm{super}\:\mathrm{fastest}\:\mathrm{prof} \\$$