Integral of (cos x)^2 - Solution 1/2x + 1/4 sin2x

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In summary, the integral of (cos x)^2 is 1/2 x + 1/4 sin 2x + C, where C is the constant of integration. To solve the integral, you can use the trigonometric identity cos^2 x = (1 + cos 2x)/2 and the power rule for integration. The constant of integration is added to the solution to represent the family of all possible solutions. The 1/2 and 1/4 coefficients in the solution are used because of the power rule and the constant multiple rule for integration. The solution can be further simplified using trigonometric identities and algebraic manipulation.
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teng125
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integral (cos x)^2??
the answer is 1/2x + 1/4 sin2x
pls help...thanx...
 
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teng125 said:
integral (cos x)^2??
the answer is 1/2x + 1/4 sin2x
pls help...thanx...
To integrate something like sin2x dx, cos2x dx, we use Power-reduction formulas, ie:
[tex]\cos ^ 2 x = \frac{1 + \cos(2x)}{2} \quad \mbox{and} \quad \sin ^ 2 x = \frac{1 - \cos(2x)}{2}[/tex]
Can you go from here?
 
  • #3
You can do this using trig identities:
[tex]\cos{2\theta}=\cos^2{\theta}-\sin^2{\theta} = 2\cos^2{\theta}-1[/tex]
Now solve for [itex]\cos^2{\theta}[/itex]:
[tex]\cos^2{\theta}=\frac{1+\cos{2\theta}}{2}[/tex]
So
[tex]\int\cos^2{x}dx=\frac{1}{2}\int 1+\cos{2x}dx[/tex]
 

FAQ: Integral of (cos x)^2 - Solution 1/2x + 1/4 sin2x

What is the integral of (cos x)^2?

The integral of (cos x)^2 is 1/2 x + 1/4 sin 2x + C, where C is the constant of integration.

How do you solve the integral of (cos x)^2?

To solve the integral of (cos x)^2, you can use the trigonometric identity cos^2 x = (1 + cos 2x)/2. Then, you can use the power rule for integration to find the integral.

What is the purpose of the constant of integration in the solution?

The constant of integration is added to the solution because when you take the derivative of the solution, the constant will disappear. It represents the family of all possible solutions to the integral.

What is the significance of the 1/2 and 1/4 coefficients in the solution?

The 1/2 coefficient is used because of the power rule for integration, which states that when taking the integral of x^n, you divide the exponent by the new exponent. In this case, the original exponent of 2 is divided by 2 to get 1. The 1/4 coefficient is used because of the constant multiple rule for integration, which states that when taking the integral of a constant times a function, you can pull the constant out and integrate the function.

Can the solution be simplified further?

Yes, the solution can be simplified further using trigonometric identities and algebraic manipulation. For example, you can use the double angle formula for sine to simplify the solution to 1/2 x + 1/4 sin x cos x + C.

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