# Thread: trig integral 1/(1-sin(x)cos(x))

1. ## trig integral 1/(1-sin(x)cos(x))

$\int_{0}^{\pi /2}{\frac{dx}{1-\sin (x)\cos (x)}}$

i saw this integral in a previous post and i'm curious on how to do it. i tried a wierstrauss substitution but that seems to make the problem extremely complicated. i also tried rewriting as ${\frac{dx}{1-{\frac{1}{2}}\sin (2x)}}$ but i'm not sure on what to do next.

what would be the best way to tackle this integral?

2. Originally Posted by oblixps
$\int_{0}^{\pi /2}{\frac{dx}{1-\sin (x)\cos (x)}}$

i saw this integral in a previous post and i'm curious on how to do it. i tried a wierstrauss substitution but that seems to make the problem extremely complicated. i also tried rewriting as ${\frac{dx}{1-{\frac{1}{2}}\sin (2x)}}$ but i'm not sure on what to do next.

what would be the best way to tackle this integral?
Make the substitution

$u = \tan{\frac{x}{2}}$ so $du = \frac{1}{2}\sec^2{\frac{x}{2}}$

and make appropriate transformations for $\sin{x}$ and $\cos{x}$.

3. yes that is the Weierstrass substitution. i tried that but it seemed to make the problem very complicated. is that the best way to do this problem?

4. Originally Posted by oblixps
yes that is the Weierstrass substitution. i tried that but it seemed to make the problem very complicated. is that the best way to do this problem?

consider $1 = \sin^2(x) + \cos^2(x)$

$\int \frac{dx}{ 1- \sin(x)\cos(x) }$

$= \int \frac{dx}{\sin^2(x) - \sin(x)\cos(x) + \cos^2(x) }$

$= \int \frac{ \sec^2(x)~dx }{ \tan^2(x) - \tan(x) + 1}$

Sub. $t = \tan(x) , dt = \sec^2(x)dx$

$I = \int_0^{\infty} \frac{dt}{t^2 - t + 1}$

5. Multiply numerator and denominator by $1 + sinxcosx$ and then substitute $cos^2x = 1 - sin^2x$ and you should be able to work your way from there.

$1 - sin^4x = (1 - sin^2x)(1 + sin^2x)$ and then use partial fractions (just an alternate way of solving).

$1 + sin^2x = (1 + isinx)(1 - isinx)$ and then use partial fractions again.

[Next time, make all of your ideas into one single post. -K.]

6. [IMG]file:///C:/Users/Guest/Desktop/properties%20of%20definite%20integrals.bmp[/IMG]
i was wondering whether some property of definite integral could be used.