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Evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{2}} \sin^2 x \cos^2 x dx $

$\int_{0}^{\pi / 2} \sin ^{2} x \cos ^{2} x d x=\frac{\pi}{16}$

Integration Techniques

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This problem is from Chapter seven section to problem number eleven of the book Cafe Louis Early Transcendental Sze eighth Edition by James Store Here we have a definite in world from syrup I over two of sine squared x Times co sign squared of X. So for this problem, since we have even powers on the sign and the co sign, it will be helpful to use these identities written on the side for Science square and coastlines where so let's apply both of these at the same time. So first we can replace science where with one minus co sign of two X all over, too. And for co sign, we have one plus cosign of two ex all over, too. So first we can pull off this one fourth out of the denominator. Pull this out in front of the integral sign and multiply these new writers together to get one minus co sign squared of two X Now for this coastline square, we could once again apply this identity over here in the blue To write this as one plus call, sign off for eggs over to so simplifying these fractions we have And now we could evaluate each of the sense of rules. Where for the second integral am I helped to use U substitution. So we have so one half becomes eggs over too. And for the second and rule, we get minus sign for eggs over four times to which is a and then we have her in points zero two pi over too. Supplying these in. We have a one over four pilot for minus sign of two pi divided by eight. And when we plug in zero, we have zero over too. Minus sign zero over eight. So sign of two pi zero. This goes away. That zero sign of zero is also zero. So we're simply left with hi over sixteen, and that's your final answer.