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Question Number 38122 by maxmathsup by imad last updated on 22/Jun/18
calculate  ∫_0 ^π     ((sinx)/( (√(1+cos^2 x))))dx
$${calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{sinx}}{\:\sqrt{\mathrm{1}+{cos}^{\mathrm{2}} {x}}}{dx} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jun/18
t=cosx   dt=−sinxdx  ∫_1 ^(−1) ((−dt)/( (√(1+t^2 ))))  ∫_(−1) ^1 (dt/( (√(1+t^2 ))))  l  =∣ln(t+(√(1+t^2 )))∣_(−1) ^1   =ln(1+(√2) )−ln(−1+(√2) )
$${t}={cosx}\:\:\:{dt}=−{sinxdx} \\ $$$$\int_{\mathrm{1}} ^{−\mathrm{1}} \frac{−{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }} \\ $$$$\int_{−\mathrm{1}} ^{\mathrm{1}} \frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }} \\ $$$${l} \\ $$$$=\mid{ln}\left({t}+\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\right)\mid_{−\mathrm{1}} ^{\mathrm{1}} \\ $$$$={ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\:\right)−{ln}\left(−\mathrm{1}+\sqrt{\mathrm{2}}\:\right) \\ $$

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