What Are Some Substitutions That Can Be Used to Solve This Integral?

In summary, the conversation discusses how to integrate the function ##\int{\sqrt{\frac{cosx - cos^3x} {1-cos^3x}}}\,dx## by using various substitutions. It is ultimately determined that the substitution ##t=\sin^{2/3}s## or ##\sqrt{1-cos^3x} = t## is the most effective in solving the integral. The final result is ##\frac{-2}{3}\sin^{-1}{(\sqrt{1-cos^3x})} + c##.
  • #1
baldbrain
236
21
Homework Statement
Evaluate $$\int{\sqrt{\frac{cosx - cos^3x} {1-cos^3x}}}\,dx$$
Relevant Equations
All the basic Integration formulae
Let I = ##\int{\sqrt{\frac{cosx - cos^3x} {1-cos^3x}}}\,dx##
I = ##\int{\sqrt{\frac{cosx(1 - cos^2x)} {1 - cos^3x}}}\,dx##
I = ##\int{\sqrt{\frac {cosx} {1 - cos^3x }}}sinx\,dx##
Substitute ##cosx = t##
Therefore, ##sinx\,dx = -dt##
So, I = ##\int{-\sqrt{\frac {t} {1 - t^3}}}\,dt##
I'm stuck here...
 
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  • #2
the integrand does not exist as a real number in ##\left [\pi/2, 3\pi/2\right ] + 2n\pi##
 
  • #3
Try the substitution ##t=\sin^{2/3}s##.
 
  • #4
BvU said:
the integrand does not exist as a real number in ##\left [\pi/2, 3\pi/2\right ] + 2n\pi##
My book says if a function is discontinuous (or doesn't exist) at a point, it's not necessary that it's anti-derivative is also discontinuous ( or doesn't exist) at that point. They've given the example of
$$\int x^{-1/3}\,dx$$ Even though the function doesn't exist at x = 0, it's anti-derivative ##\frac {x^{2/3}} {2/3} + c## is continuous at x = 0
 
  • #5
MathematicalPhysicist said:
Try the substitution ##t=\sin^{2/3}s##.
Yes, that seems to do it. :smile:
##dt=(\frac{-2}{3})(\sin^{-1/3}s)(\cos{s})ds##. Also, ##\sqrt{t} = sin^{1/3}s## & ##\sqrt{1 - t^3} = cos s##
Hence, I = ##\int{\frac{-2}{3}\frac{(sin^{1/3}s)(coss)} {(coss)(sin^{1/3}s)}}\,ds##
I = ##\int{\frac{-2}{3}}\,ds## = ##\frac{-2}{3}s + c##
I = ##\frac{-2}{3}\sin^{-1}{(cos^{3/2}x)} + c##
Thanks
 
  • #6
Alternatively, the substitution ##\sqrt{1-cos^3x} = t## also works.
That yields
I = ##\frac{2}{3}\sin^{-1}{(\sqrt{1-cos^3x})} + c##
 

Related to What Are Some Substitutions That Can Be Used to Solve This Integral?

1. What is a "Head-Scratching Integral"?

A "Head-Scratching Integral" is a term used to describe a particularly difficult and complex mathematical integral that requires a lot of thought and problem-solving to solve.

2. How is a "Head-Scratching Integral" different from a regular integral?

A "Head-Scratching Integral" is typically much more challenging and requires more advanced mathematical techniques to solve compared to a regular integral. It often involves complex functions or multiple variables.

3. Why are "Head-Scratching Integrals" important in science?

"Head-Scratching Integrals" are important in science because they often arise in real-world problems and require creative thinking and problem-solving skills to find a solution. They can also lead to new discoveries and advancements in various fields of science.

4. How do scientists approach solving a "Head-Scratching Integral"?

Scientists approach solving a "Head-Scratching Integral" by first understanding the problem and its context, then breaking it down into smaller, more manageable parts. They may also use various mathematical techniques and tools, such as integration by parts or substitution, to help them find a solution.

5. Are there any tips for solving "Head-Scratching Integrals"?

Some tips for solving "Head-Scratching Integrals" include practicing regularly, breaking down the problem into smaller parts, and using various mathematical techniques. It is also helpful to have a good understanding of the properties of integrals and to be familiar with common mathematical functions and their derivatives.

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