How do you find the integral of #int cos x * sin x dx#?
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To find the integral of ∫cos(x) * sin(x) dx, you can use integration by parts. Let u = cos(x) and dv = sin(x) dx. Then, differentiate u to get du and integrate dv to get v. After that, apply the integration by parts formula:
∫u dv = uv - ∫v du
Substitute the values of u, dv, du, and v into the formula and solve the integral. This yields:
∫cos(x) * sin(x) dx = -cos(x) * cos(x) - ∫(-cos(x) * (-sin(x))) dx
Now, integrate the remaining term:
∫(-cos(x) * (-sin(x))) dx = ∫cos(x) * sin(x) dx
So, you get:
∫cos(x) * sin(x) dx = -cos^2(x) - ∫cos(x) * sin(x) dx
Rearrange the equation to solve for the integral:
2∫cos(x) * sin(x) dx = -cos^2(x)
Now, solve for the integral:
∫cos(x) * sin(x) dx = -1/2 * cos^2(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.
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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