Integrate (sin x)^3: Simplify w/o Parts

  • Thread starter Chadlee88
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In summary, when integrating (sin x)^3, we can use the substitution u = cos(x) and change all sine functions to cosine functions using the Pythagorean Identity. This is the general method for integrating odd powers of the sine function.
  • #1
Chadlee88
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Could someone please help me integrate (sin x)^3. Can i use any simpler method asides from integration by parts.??
 
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  • #2
[tex] \int \sin^{3} x \; dx = \int (1-\cos^{2}x)\sin x \; dx [/tex].

Let [tex] u = \cos x [/tex]
 
  • #3
Why do we have so many people who think differentiation and integeration are pre-calculus?
 
  • #4
yeh its kind of the crux of calc rly
 
  • #5
Generally, when the power of the sine function is odd, we use the substitution u = cos(x), and change all sine functions to cosine functions by using the Pythagorean Identity : sin2x + cos2x = 1.
When the power of the cosine function is odd, we use the substitution u = sin(x), and change all cosine functions to sine functions by using the Pythagorean Identity : sin2x + cos2x = 1.
When both powers are even, we use the Power-Reduction Formulae. :)
And in your problem, the power of sine is odd, hence, we use the substitution: u = cos(x)
 

Related to Integrate (sin x)^3: Simplify w/o Parts

What is the formula for integrating (sin x)^3?

The formula for integrating (sin x)^3 is:

∫ (sin x)^3 dx = ∫ sin^2 x * sin x dx

Can (sin x)^3 be simplified without using integration by parts?

Yes, (sin x)^3 can be simplified without using integration by parts by using the trigonometric identity sin^2 x = (1 - cos 2x)/2.

What is the general approach for simplifying (sin x)^3 without integration by parts?

The general approach for simplifying (sin x)^3 without integration by parts is to use the trigonometric identity sin^2 x = (1 - cos 2x)/2 to rewrite the integral as:

∫ (sin x)^3 dx = ∫ (1 - cos 2x)/2 * sin x dx

How do you integrate the resulting integral after simplifying (sin x)^3?

After simplifying (sin x)^3, the resulting integral can be solved by applying the power rule, integration by substitution, or integration by parts.

What is the final solution for integrating (sin x)^3 without using integration by parts?

The final solution for integrating (sin x)^3 without using integration by parts is:

∫ (sin x)^3 dx = -cos x - (1/4) * cos 3x + C

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