# How do you integrate #int cos^3thetasqrt(sintheta)#?

by inspection

which can be simplified as required

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To integrate ( \int \cos^3(\theta) \sqrt{\sin(\theta)} ), we can use the trigonometric identity ( \cos^2(\theta) = 1 - \sin^2(\theta) ) to express ( \cos^3(\theta) ) in terms of ( \sin(\theta) ). Then, we can perform a substitution to simplify the integral.

Let: [ u = \sin(\theta) ] [ du = \cos(\theta) , d\theta ]

Now, substitute ( u = \sin(\theta) ) and ( du = \cos(\theta) , d\theta ): [ \int \cos^3(\theta) \sqrt{\sin(\theta)} , d\theta = \int (1 - \sin^2(\theta)) \sqrt{u} , du ]

This simplifies the integral to: [ \int (1 - u^2) \sqrt{u} , du ]

Now, you can integrate ( (1 - u^2) \sqrt{u} ) with respect to ( u ). This can be done by expanding ( (1 - u^2) ) and using techniques like substitution or integration by parts.

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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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