Integral of (cos(x))^2 in the hard way

In summary, the conversation discusses using integration by parts to solve the integral of cos(x) squared without using the trigonometric identity. The speaker suggests rewriting it as cos(x)*cos(x) and using the fact that sin^2 = 1 – cos^2. They also provide an example of how they would approach the problem using IBP.
  • #1
Yankel
395
0
Hello guys

I am trying to solve the integral of cos(x) squared, i.e. (cos(x))^2, but not using the trigonometric identity of this function, but using integration by parts.

I tried turning it into 1*(cos(x))^2, but I didn't go too far, maybe did something wrong.

Am I on the right direction ?

cheers
 
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  • #2
Yankel said:
Hello guys

I am trying to solve the integral of cos(x) squared, i.e. (cos(x))^2, but not using the trigonometric identity of this function, but using integration by parts.

I tried turning it into 1*(cos(x))^2, but I didn't go too far, maybe did something wrong.

Am I on the right direction ?
You would do better to write it as cos(x)*cos(x). After integrating by parts, use the fact that sin^2 = 1 – cos^2.
 
  • #3
If I were going to tackle this using IBP, I would let:

\(\displaystyle u=\cos(x)\,\therefore\,du=-\sin(x)\,dx\)

\(\displaystyle dv=\cos(x)\,dx\,\therefore\,v=\sin(x)\)

What do you find?
 

FAQ: Integral of (cos(x))^2 in the hard way

1) What is the formula for calculating the integral of (cos(x))^2 in the hard way?

The formula for calculating the integral of (cos(x))^2 in the hard way is:
∫ (cos(x))^2 dx = ∫ ½(1 + cos(2x)) dx

2) What is the step-by-step process for solving the integral of (cos(x))^2 in the hard way?

The step-by-step process for solving the integral of (cos(x))^2 in the hard way is:
1) Use the double angle formula: cos(2x) = 1 - 2sin^2(x)
2) Substitute cos(2x) with 1 - 2sin^2(x)
3) Distribute the ½ into the integral
4) Integrate each term separately
5) Simplify and combine terms

3) Why is it called the "hard way" to solve the integral of (cos(x))^2?

It is called the "hard way" because it requires using a double angle formula and integrating each term separately, which can be more time-consuming and complex compared to other methods such as integration by substitution or integration by parts.

4) Can the integral of (cos(x))^2 be solved using other methods?

Yes, there are other methods for solving the integral of (cos(x))^2 such as integration by substitution or integration by parts. These methods may be simpler and more efficient compared to the "hard way".

5) What are some practical applications of calculating the integral of (cos(x))^2?

The integral of (cos(x))^2 has various applications in physics, engineering, and mathematics. For example, it can be used to calculate the work done by a force acting at an angle, or to find the average value of a periodic function over a given interval. It is also used in the derivation of trigonometric identities and in solving differential equations.

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