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Problem 41 Hard Difficulty

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.

$ \displaystyle \int \cos^4 x\ dx $


$$\frac{1}{4} \cos ^{3} x \sin x+\frac{3}{8} x+\frac{3}{8} \sin x \cos x+C$$


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Video Transcript

Okay. This question wants us to find this anti derivative using technology and then compare that to what we get using tables. So computer algebra system gives us this answer which condenses everything nicely into double and quadruple angles. Front. Our table appears to have expanded something out. So let's try and turn our CASS answer into our table. Answer. So really, it looks like what we want to dio is get rid of this sign of four X So we're going to use a trick identity for this. So remember sign of two X is to sign ex co sign X and Co sign of two X equals co sign Squared Acts minus Sign scored X. So that means that sign of four X is equal to to sign two ex co sign of two acts, which equals two times to sign. Ex co sign X Times Co sign Squared X minus Sign scored Axe And then, if we plug that in over over here will see that this simplifies to creates X plus Sign two x over four plus using those two identities. Four. Sign ex co sign Q Dax minus four. Sign Cubed Co Cenex over 32 stone. So now let's see what we can do with this new term. So our table answer has a coastline cubed in it, so we're gonna want to get rid of the sign Cube part so sine X coastline. Cute Dex over eight minus well, sine squared times sign Ex co sign X. So then we can simplify this again. Minus well, sine squared axes a one minus co sign squared ites. And now we can expand things out again and get 3/8 x plus Sign to x over four minus sign Ex co sign X Over eight Minus sorry plus sign Ex co signed Cute Dex over a plus Sign ex co sign Q Backs over a and now we can simplify this as 3/8 x minus. Signed two x over four minus signed to x over 16 plus sign ex co sign Cube Dex over four. And that's just combining common denominators. So now this should be a plus. So now we just need to combine these two signed two ex terms and we get our final answer because 4/16 miners 1/16 is 3/16 and this matches what our table told us. The answer should be 3/8 1 4th 3/16 all with positive science

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