How do you find the antiderivative of cos2(x) ?

Marla Payton

Marla Payton

Answered

2021-12-26

How do you find the antiderivative of cos2(x) ?

Answer & Explanation

Samantha Brown

Samantha Brown

Expert

2021-12-27Added 35 answers

The secret to figuring out this integral is to use an identity, in this case, the cosine double-angle identity.
Since cos(2x)=cos2(x)sin2(x). We can rewrite this using the Pythagorean Identity to say that cos(2x)=2cos2(x)1. Solving this for cos2(x) shows us that cos2(x)=cos(2x)+12 
Thus: 
cos2(x) dx =12cos(2x)+1 dx  
Now that we have separated this, we can locate the antiderivative.
=12cos(2x) dx +121 dx  
=142cos(2x) dx +12x 
=14sin(2x)+12x+C

ol3i4c5s4hr

ol3i4c5s4hr

Expert

2021-12-28Added 48 answers

We can't just integrate cos2(x) as it is, so we want to change it into another form, which we can easily do using trig identities.
Recall the double angle formula: cos(2x)=cos2(x)sin2(x). We also know the trig identity sin2(x)+cos2(x)=1, so combining these we get the equation cos(2x)=2cos2(x)1.
Now we can rearrange this to give: cos2(x)=1+cos(2x)2
So we have an equation which gives cos2(x) in a nicer form which we can easily integrate using the reverse chain rule.
This eventually gives us an answer of x2+sin(2x)4+C
nick1337

nick1337

Expert

2022-01-08Added 573 answers

Th antiderivative is pretty much the same as the integral, except it;s more general,
so I'll do the indefinite integral.
cos2ddxAn identify for cos2x is:cos2x=1+cos(2x)2121+cos(2x)dxSince ddx[sin(2x)]=2(2x) cos(2x)dx=12sin(2x)sin(2x)=2sinxcosxso12sin(2x)=sinxcosx12[x+12sin(2x)]+C=x2+14sin(2x)+C=x2+12sinxcosx+C

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