How do you find the integral of #int sin^6(x) cos^3(x) dx#?
We can integrate by substitution since sine and cosine are derivatives of one another (up to a mminus sign).
Complete the substitution, then integrate, expand, and resultant polynomial.
To complete the substitution, reverse it.
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To find the integral of ( \int \sin^6(x) \cos^3(x) , dx ), you can use the method of integration by parts. Integration by parts states that ( \int u , dv = uv - \int v , du ), where ( u ) and ( v ) are differentiable functions of ( x ).
In this case, let ( u = \sin^6(x) ) and ( dv = \cos^3(x) , dx ). Then, differentiate ( u ) to find ( du ), and integrate ( dv ) to find ( v ).
Differentiating ( u ): [ du = 6\sin^5(x) \cos(x) , dx ]
Integrating ( dv ): [ v = \int \cos^3(x) , dx ] [ = \int \cos^2(x) \cos(x) , dx ] [ = \int (1 - \sin^2(x)) \cos(x) , dx ] [ = \int (\cos(x) - \sin^2(x) \cos(x)) , dx ] [ = \sin(x) - \int \sin^2(x) \cos(x) , dx ]
Now, we have ( u ), ( du ), ( v ), and ( dv ), so we can apply integration by parts:
[ \int \sin^6(x) \cos^3(x) , dx = uv - \int v , du ] [ = \sin^6(x) \sin(x) - \int (\sin(x) - \int \sin^2(x) \cos(x) , dx) (6\sin^5(x) \cos(x)) , dx ]
This integral is now reduced to the integral of ( \sin^2(x) \cos(x) ), which can be solved using integration by parts again, or through trigonometric identities.
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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.
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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