# What is the integral of #int cos^3 (x)dx# from 0 to pi/2?

substitute:

so:

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To find the integral of (\int \cos^3(x) , dx) from (0) to (\frac{\pi}{2}), we can use the trigonometric identity ( \cos^3(x) = (\cos(x))^3 ) and then perform a substitution.

Let ( u = \sin(x) ), then ( du = \cos(x) , dx ).

So, the integral becomes:

[ \int \cos^3(x) , dx = \int (\cos(x))^3 , dx = \int \cos(x) \cdot \cos^2(x) , dx = \int (1 - \sin^2(x)) , du ]

This simplifies to:

[ \int (1 - u^2) , du = \int (1 - u^2) , du = u - \frac{u^3}{3} + C ]

Now, we need to substitute back (u = \sin(x)):

[ = \sin(x) - \frac{\sin^3(x)}{3} + C ]

Now, evaluate this expression from (0) to (\frac{\pi}{2}):

[ \left[ \sin(x) - \frac{\sin^3(x)}{3} \right]_0^{\frac{\pi}{2}} = \left[ \sin\left(\frac{\pi}{2}\right) - \frac{\sin^3\left(\frac{\pi}{2}\right)}{3} \right] - \left[ \sin(0) - \frac{\sin^3(0)}{3} \right] ]

[ = \left[ 1 - \frac{1}{3} \right] - \left[ 0 - 0 \right] = \frac{2}{3} ]

So, the integral of (\cos^3(x)) from (0) to (\frac{\pi}{2}) is (\frac{2}{3}).

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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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