How do you find the integral of #(Sin2x) / (1 + cos^2x) dx#?
Make use of trig identities...
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Second method
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To find the integral of (\frac{\sin(2x)}{1 + \cos^2(x)}) with respect to (x), use the substitution method. Let (u = \cos(x)), then (du = -\sin(x) dx).
So, (\sin(2x)dx = -\frac{1}{2}du).
The integral becomes (-\frac{1}{2}\int \frac{du}{1 + u^2}).
This is a standard integral, which evaluates to (-\frac{1}{2}\arctan(u) + C), where (C) is the constant of integration.
Now, substitute back (u = \cos(x)) to get the final answer:
(-\frac{1}{2}\arctan(\cos(x)) + C).
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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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