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Question Number 144214 by Mathspace last updated on 23/Jun/21 | ||
$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{4}\pi} \:\:\frac{{sinx}}{\left(\mathrm{3}+{cosx}\right)^{\mathrm{2}} }{dx} \\ $$ | ||
Answered by bemath last updated on 23/Jun/21 | ||
$$=−\underset{\mathrm{0}} {\overset{\mathrm{4}\pi} {\int}}\:\frac{\mathrm{d}\left(\mathrm{3}+\mathrm{cos}\:\mathrm{x}\right)}{\left(\mathrm{3}+\mathrm{cos}\:\mathrm{x}\right)^{\mathrm{2}} } \\ $$$$=−\left[\:−\frac{\mathrm{1}}{\mathrm{3}+\mathrm{cos}\:\mathrm{x}}\:\right]_{\mathrm{0}} ^{\mathrm{4}\pi} \\ $$$$=\:\left[\frac{\mathrm{1}}{\mathrm{3}+\mathrm{cos}\:\mathrm{4}\pi}\right]−\left[\frac{\mathrm{1}}{\mathrm{3}+\mathrm{cos}\:\mathrm{0}}\:\right]\:=\mathrm{0} \\ $$ | ||