Help with this Definite Integral

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SUMMARY

The discussion focuses on integrating the function f(x) = |(sin X)^3 * (cos X)^15| over the interval [0, 2π]. The user simplifies the integral and divides it into four segments of π/2 each. The main question arises regarding the second quadrant interval [π/2, π], specifically how to express f(x) without the absolute value, considering that sine is positive and cosine is negative in this quadrant. The conclusion reached is that the user correctly identifies the behavior of the function in this interval.

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  • Understanding of definite integrals
  • Knowledge of trigonometric functions and their properties
  • Familiarity with the concept of absolute value in mathematical functions
  • Ability to break down integrals into sub-intervals
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  • Study integration techniques for trigonometric functions
  • Learn about the properties of absolute values in integrals
  • Explore the use of symmetry in definite integrals
  • Practice solving integrals involving powers of sine and cosine
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Students preparing for calculus exams, mathematics educators, and anyone looking to deepen their understanding of integration techniques involving trigonometric functions.

chuachinghong
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Well as I am practising for my coming test, I encountered this question:

Integrate
f(x) = absolute [(sin X)^3 * (cos X)^15]dx
within the interval of [0,2pi]


I tried simplifying this integral into...


f(x) = absolute[ ((cos X)^15)*(sin X) -((cos X)^17)*(sin X))] dx
within the interval of [0,2pi]

Also, I break up the integral into 4 parts of pi/2 each i.e [0,pi/2], [pi/2,pi], [pi,3pi/2] and [3pi/2, 2pi]

My question is, when I am calculating the interval of the Second Quadrant, which is from [pi/2, pi]. What should my f(x) look like after taking out the absolute sign?


Hope you can help me with this. Thank you :smile:
 
Last edited:
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sine is positive, cosine is negative in the second quadrant, and your powers are odd, so...?
 
Oh ok I got it. Thanks for your help. @,@
 

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