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 nuggetprime 2007-06-26 19:11

Many CAS not able to solve integral

Hello,
I tried(with wolfram integrator, mupad & derive) to solve this integral:

SIN(x)·(1 - COS(x))·
!!! ANGLES IN DEGREE!!!

None of the three Computer Algebra Systems were able to do this correct(i set angle options). Can anybody help me?

Thanks,
Nuggetprime

 alpertron 2007-06-26 22:04

Let x be the angle in degrees, so the angle in radians is:

$$\frac{\pi}{180}x$$

So you want to integrate:

$$\sin \left(\frac{\pi}{180}x\right)\left(1-cos(\frac{\pi}{180}x)\right)\,=\, \sin \left(\frac{\pi}{180}x\right)-\sin \left(\frac{\pi}{180}x\right)\,cos(\frac{\pi}{180}x)$$

using standard sin and cos functions in radian.

Since $$\sin(2x) = 2\sin x \cos x$$ we get:

$$\sin \left(\frac{\pi}{180}x\right)-\frac{1}{2}\sin \left(\frac{\pi}{90}x\right)$$

And its integral is:

$$-\frac{180}{\pi}\cos \left(\frac{\pi}{180}x\right)+\frac{45}{\pi}\cos \left(\frac{\pi}{90}x\right)$$

So the result in degrees is (using cosines in degrees):

$$-\frac{180}{\pi}\cos (x)+\frac{45}{\pi}\cos (2x)$$

I don't know the solution given by these packages but the difference between their results and the one I gave above should be a constant. Please notice that there are many ways to represent trigonometric identities so it is possible that two very different looking expressions are equal.

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