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### Integral of Cos^2

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Date: 2/6/96 at 18:26:12
From: Anonymous
Subject: Anti-derivative for Calculus

I can't seem to find the Anti-derivative of (cos X)^2.  I have tried to
find my own solution, which is: (-(cos X)^3)/(3sin x).  I would like
```

```
Date: 6/13/96 at 13:58:39
From: Doctor Charles
Subject: Re: Anti-derivative for Calculus

Well:

d/dx[(-(cos x)^3)/(3 sin x)] = (9(sin(x)cos(x))^2 -3
(cos(x))^4))/(9(sin(x)^2)

= (cos(x))^2 * (1 - (1/3) (cot(x))^2)

so I am afraid that your solution doesn't work. It looks as though
you have used a substitution without doing it rigorously enough.
Here's how it should be done.

cos(2x) = (cos(x))^2 - (sin(x))^2

cos(2x) = 2 cos(x)^2 - 1
(cos(x))^2 = 1/2 * (cos(2x) + 1)

if intgl() stands for the integral (anti-derivative) of a function then:

intgl(cos(2x)) = 1/2 * sin(2x) + 2C
so
intgl((cos(x))^2) = 1/4 sin(2x) + 1/2 x + C

-Doctor Charles,  The Math Forum

```
Associated Topics:
High School Calculus

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