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

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.
Could someone please help me integrate (sin x)^3. Can i use any simpler method asides from integration by parts.??

$$\int \sin^{3} x \; dx = \int (1-\cos^{2}x)\sin x \; dx$$.

Let $$u = \cos x$$

Why do we have so many people who think differentiation and integeration are pre-calculus?

yeh its kind of the crux of calc rly

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)

## 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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