# How do you find the integral of #(sinx)^3 dx#?

Refer to explanation

We are aware of that

#sin 3x = 3sinx - 4 (sin x)^3=>4(sinx)^3=3sinx-sin3x=> (sinx)^3=3/4*sinx-1/4sin3x #

Thus, we have that

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To find the integral of ((\sin{x})^3 , dx), we can use a trigonometric identity to rewrite it in terms of a different trigonometric function. We'll use the identity:

((\sin{x})^3 = (\sin{x})(1 - \cos^2{x}))

Then, we'll make a substitution: let (u = \cos{x}), so (du = -\sin{x} , dx). The integral becomes:

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

(= -\int (1 - u^2) , du)

(= -\int (1 - u^2) , du)

(= -\int (1 , du - u^2 , du))

(= -\left(u - \frac{u^3}{3}\right) + C)

Now, we'll substitute back (u = \cos{x}):

(= -\left(\cos{x} - \frac{\cos^3{x}}{3}\right) + C)

So, the integral of ((\sin{x})^3 , dx) is (-\cos{x} + \frac{\cos^3{x}}{3} + C), where (C) is the constant of integration.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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