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Question Number 144214 by Mathspace last updated on 23/Jun/21

calculate ∫_0 ^(4π)   ((sinx)/((3+cosx)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{4}\pi} \:\:\frac{{sinx}}{\left(\mathrm{3}+{cosx}\right)^{\mathrm{2}} }{dx} \\ $$

Answered by bemath last updated on 23/Jun/21

=−∫_0 ^(4π)  ((d(3+cos x))/((3+cos x)^2 ))  =−[ −(1/(3+cos x)) ]_0 ^(4π)   = [(1/(3+cos 4π))]−[(1/(3+cos 0)) ] =0

$$=−\underset{\mathrm{0}} {\overset{\mathrm{4}\pi} {\int}}\:\frac{\mathrm{d}\left(\mathrm{3}+\mathrm{cos}\:\mathrm{x}\right)}{\left(\mathrm{3}+\mathrm{cos}\:\mathrm{x}\right)^{\mathrm{2}} } \\ $$$$=−\left[\:−\frac{\mathrm{1}}{\mathrm{3}+\mathrm{cos}\:\mathrm{x}}\:\right]_{\mathrm{0}} ^{\mathrm{4}\pi} \\ $$$$=\:\left[\frac{\mathrm{1}}{\mathrm{3}+\mathrm{cos}\:\mathrm{4}\pi}\right]−\left[\frac{\mathrm{1}}{\mathrm{3}+\mathrm{cos}\:\mathrm{0}}\:\right]\:=\mathrm{0} \\ $$

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