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Question Number 61147 by malwaan last updated on 29/May/19
prove  ∫((1+cos x)/(1−cos x))dx=−2cot (x/2)−x+c
$$\boldsymbol{{prove}} \\ $$$$\int\frac{\mathrm{1}+{cos}\:{x}}{\mathrm{1}−{cos}\:{x}}{dx}=−\mathrm{2}{cot}\:\frac{{x}}{\mathrm{2}}−{x}+{c} \\ $$$$ \\ $$
Answered by tanmay last updated on 29/May/19
∫((2cos^2 (x/2))/(2sin^2 (x/2)))dx  ∫(cosec^2 (x/2)−1)dx  =((−cot((x/2)))/(1/2))−x+c  =−2cot((x/2))−x+c
$$\int\frac{\mathrm{2}{cos}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{\mathrm{2}{sin}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}}{dx} \\ $$$$\int\left({cosec}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}−\mathrm{1}\right){dx} \\ $$$$=\frac{−{cot}\left(\frac{{x}}{\mathrm{2}}\right)}{\frac{\mathrm{1}}{\mathrm{2}}}−{x}+{c} \\ $$$$=−\mathrm{2}{cot}\left(\frac{{x}}{\mathrm{2}}\right)−{x}+{c} \\ $$
Commented by malwaan last updated on 29/May/19
thank you sir  can you solve it by trig. sub. ?
$${thank}\:{you}\:{sir} \\ $$$${can}\:{you}\:{solve}\:{it}\:{by}\:{trig}.\:{sub}.\:? \\ $$

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