## siddarth95 Group Title Help prove this integral one year ago one year ago

1. siddarth95 Group Title

$\int\limits_{0}^{\pi/2}\frac{ dx }{ \sin x + \cos x } = \frac{ 1 }{ \sqrt{2} } \ln \ \frac{ \sqrt{2}+1 }{ \sqrt{2}-1 }$

let u=sin x +cos x du=cosx-sinx dx

$\int \frac{1}{ \cos x+\sin x}dx$ multiply by cos x-sin x both numerator and denominator

$\int \frac{\cos x-\sin x}{\cos^2x-\sin^2x} dx=\int \frac{\cancel{\cos x-\sin x}du}{\cos 2x}\frac{du}{\cancel{\cos x-\sin x}}$

$\int \frac{du}{\cos 2x}=\int \sec 2x=\ln|\sec 2x+\tan 2x|$/2

error i never substituted u

7. Meepi Group Title

I don't have time to put an explanation here, but use the substitution u = tan(x/2): http://www-math.mit.edu/~djk/18_01/chapter24/section03.html

8. Meepi Group Title

9. experimentX Group Title

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10. siddarth95 Group Title

its all good .. thanks for the responses :)

11. experimentX Group Title

there's one another way to do it .. change all trigs into half angles, and change sines and cosines into tan and sec .. you should end up something like|dw:1362841238898:dw|