# What is #int sin^3x/cos^6x dx#?

where

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To integrate (\frac{{\sin^3(x)}}{{\cos^6(x)}}) with respect to (x), use the substitution (u = \cos(x)). This will change the integral into one involving only (u). Then use trigonometric identities to simplify the resulting integral. The steps are as follows:

- Let (u = \cos(x)), then (du = -\sin(x)dx).
- Substitute (-\sin(x)dx = du) and (\sin^2(x) = 1 - \cos^2(x)) into the integral.
- Rewrite the integral entirely in terms of (u).
- Simplify the integral using trigonometric identities.
- Integrate the resulting expression with respect to (u).
- Finally, resubstitute (u = \cos(x)) back into the expression.

The final result is:

[ \int \frac{{\sin^3(x)}}{{\cos^6(x)}} dx = \frac{{-\sin^2(x)}}{5\cos^5(x)} + C ]

where (C) is the constant of integration.

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To integrate ( \frac{{\sin^3(x)}}{{\cos^6(x)}} ) with respect to ( x ), you can use the trigonometric identity ( \sin^2(x) = 1 - \cos^2(x) ) to rewrite the integral in terms of ( \cos(x) ). Then make a substitution to simplify the integral.

Here's the solution:

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

Now substitute ( u ) and ( du ) into the integral:

[ \int \frac{{\sin^3(x)}}{{\cos^6(x)}} , dx = -\int \frac{{\sin^2(x) \sin(x)}}{{\cos^6(x)}} , dx ]

Using the trigonometric identity ( \sin^2(x) = 1 - \cos^2(x) ):

[ = -\int \frac{{(1 - \cos^2(x)) \sin(x)}}{{\cos^6(x)}} , dx ]

[ = -\int \frac{{(1 - u^2)(-du)}}{{u^6}} ]

[ = \int \frac{{(1 - u^2)}}{{u^6}} , du ]

[ = \int \left( u^{-6} - u^{-4} \right) , du ]

[ = \frac{{u^{-5}}}{{-5}} - \frac{{u^{-3}}}{{-3}} + C ]

[ = -\frac{1}{5u^5} + \frac{1}{3u^3} + C ]

Finally, substitute ( u = \cos(x) ) back:

[ = -\frac{1}{5\cos^5(x)} + \frac{1}{3\cos^3(x)} + C ]

So, the integral of ( \frac{{\sin^3(x)}}{{\cos^6(x)}} ) with respect to ( x ) is ( -\frac{1}{5\cos^5(x)} + \frac{1}{3\cos^3(x)} + 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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