How do you integrate #sin^5 (x) * cos^3 (x)#?
Use one of the substitutions:
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To integrate ( \sin^5(x) \cdot \cos^3(x) ), you can use trigonometric identities to simplify the expression, then perform integration by substitution. One common approach is to use the powerreduction formulas to express the powers of sine and cosine in terms of multiples of lower powers. After simplification, you can perform integration more easily.

Use the powerreduction formulas: [ \sin^2(x) = \frac{1  \cos(2x)}{2} ] [ \cos^2(x) = \frac{1 + \cos(2x)}{2} ]

Apply these formulas repeatedly to express ( \sin^5(x) ) and ( \cos^3(x) ) in terms of ( \sin(x) ) and ( \cos(x) ).

After simplifying, you will have an expression that involves powers of ( \sin(x) ) and ( \cos(x) ).

Use a substitution where ( u = \sin(x) ) or ( u = \cos(x) ), and ( du ) will be expressed in terms of ( dx ).

Perform integration using the substitution method.

Finally, substitute back the expression for ( u ) in terms of ( x ) to get the final result.
This process will yield the integrated expression for ( \sin^5(x) \cdot \cos^3(x) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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