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Problem 29 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int_0^\pi x \sin x \cos x dx $

Answer

$$
-\frac{\pi}{4}
$$

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Video Transcript

to evaluate the definite integral from 0 to Pi of X times Sine of X times Cosine of X. Dx. We know that sign of two X. This is equal to to sign up X times Cosine of X. And so 1/2 sine of two x. This is just sine of X times go sign of X. Therefore we can read our integral into the integral from 0 to Pi. Ah X times 1/2. sign up two X. Dx and we can Right the constant out. We have 1/2 Integral from 0 to Pi of X times sine of two X dx. From here we want to apply integration by parts. We let U equal to X. And DV Sign of two x. DX so that the U. S. D. X. And we would be Negative co sign of two x over two. So from here we have one half times U times V. That's negative x times go sign up to X over two minus the integral of V times do you? That's negative co sign of two X. Over to the X. Simplifying that we have one half times negative X times go Sign up to X over two Plus we have 1/2 integral of course and have two X. That's just sign of two X over two. And this will be evaluated from zero to pi. And so from here we have negative X. Co sign up to X over four Plus sign of two x Over eight. This evaluated from 0 to Pi And if excess spy we have negative pi close enough to buy Over four plus, sign up to buy over eight. This minus when X0, we have zero Plus sine of 0/8. This is just zero and so we have also. This one is zero and co sign of two pi is one. So the value of the integral would be Negative by over four.