To integrate sin^2x cos^2x, additionally written together ∫cos2x sin2x dx, sin squared x cos squared x, sin^2(x) cos^2(x), and (sin x)^2 (cos x)^2, we begin by using traditional trig identities to to change the form.
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We start by utilizing the Pythagorean trig identity and rearrange it for cos squared x to make expression <1>. Us recall the twin angle trig identity and also rearrange it because that sin squared x to make expression <2>. Us then instead of expression <2> right into expression <1> to make expression <4>, which is for cos squared x.
We repeat the previous measures using the Pythagorean and dual angle trig identities, except this time we develop an expression because that sin squared x.
Hence we deserve to rewrite sin^2x cos^2x in a new type that means the exact same thing.
We emphasis on multiplying the brackets, and also therefore relocate the fraction out that the way.
We view that the cos2x state cancel out thereby simplifying the expression.
Hence, this is the simplified expression. Periodically in mathematics, friend crawl your way out the a hole, just to autumn into another. We still have a cos squared hatchet that will certainly pose some difficulties.
We remind expression <4>, which us made previously, as displayed above.
If we multiply the angle by 2 ~ above both sides, climate the equation continues to be balanced, however, it provides us cos squared 2x because that substitution.
I choose to have it written like this. We have the right to now substitute for cos squared 2x in our previous equation <5> as shown below.
At this stage, us substitute and also simplify.
We recall our initial integration problem, and also now we have the right to write that in another type as displayed on the RHS.
We can integrate each term individually as presented on the RHS. Although the very first term top top the RHS, which is a constant, is an easy to integrate, the 2nd term will require some much more thinking. For this reason we emphasis our fist on completely the second term together shown listed below in red.
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We emphasis on complete cos4x and get the answer, as presented in red. We can now instead of this part of the solution right into our initial integration trouble as shown below.