Line integral of sin cos function

In summary, the given problem involves finding the integral of (sin^4(x) + cos^4(x))^.5 dx. Using the equation sin^2(x) = (1 - cos^2(x)), we can rewrite the equation as cos^4(x) + {1 - cos^2(x)}^2 = 2cos^4(x) - 2cos^2(x) + 1. Substituting u = cos^2(x), we get the integral of (2u^2 - 2u + 1)^.5 dx. However, the dx term is missing and should be written in the form f(u)du. Another approach could be using the identity (cos2x
  • #1
jbowers9
89
1

Homework Statement



integral (sin^4(x) + cos^4(x))^.5 dx

Homework Equations



sin^2(x) = (1 - cos^2(x))

The Attempt at a Solution



cos^4(x) + {1 - cos^2(x)}^2 = 2cos^4(x) -2cos^2(x) + 1
subst u = cos^2(x)
integral (2u^2 - 2u +1)^.5
 
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  • #2
hi jbowers9! :smile:

(have a square-root: √ and try using the X2 icon just above the Reply box :wink:)
jbowers9 said:
cos^4(x) + {1 - cos^2(x)}^2 = 2cos^4(x) -2cos^2(x) + 1
subst u = cos^2(x)
integral (2u^2 - 2u +1)^.5

no, you've missed out the dx, which you need to write in the form f(u)du :redface:

(you could try using (cos2x + sin2x)2, but i don't see how to finish that :frown:)
 

FAQ: Line integral of sin cos function

What is the definition of a line integral of sin cos function?

The line integral of sin cos function is a mathematical concept used to calculate the total value of a two-dimensional function along a specific path or curve. It involves taking the integral of the function over the specified path or curve, which can be represented by parameterizing the curve with respect to a single variable.

How is the line integral of sin cos function related to vector calculus?

The line integral of sin cos function is closely related to vector calculus, as it involves integrating a vector field along a path. This concept is used in various applications, such as calculating work done by a force field or finding the flux of a vector field through a surface.

Can the line integral of sin cos function be evaluated using any method?

Yes, the line integral of sin cos function can be evaluated using various methods, such as the fundamental theorem of calculus, Green's theorem, or Stokes' theorem. The choice of method depends on the specific problem and the given conditions.

What is the significance of the orientation of the path in the line integral of sin cos function?

The orientation of the path is crucial in the line integral of sin cos function, as it determines the direction in which the integral is evaluated. Changing the orientation of the path can result in a different value for the integral. It is important to specify the correct orientation when setting up the integral.

Can the line integral of sin cos function be applied in real-world scenarios?

Yes, the line integral of sin cos function has many practical applications in physics, engineering, and other fields. For example, it can be used to calculate the work done by a force in moving an object along a curved path, or to determine the flow of a fluid through a curved pipe.

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