Learn How To Integrate Sin(X) With -Cos(X) + C Explained

∫sin(x)dx

-cosx + c

∫sin(x)dx = -cos(x) + C

where C is the constant of integration.

To derive this result, we can use integration by substitution. Let u = cos(x), then du/dx = -sin(x) dx.

Substituting u and du into the integral, we get:

∫sin(x)dx = -∫du/u

Integrating -1/u with respect to u, we get -ln(|u|) + C.

Substituting u = cos(x) back in, we get:

∫sin(x)dx = -ln(|cos(x)|) + C

We can simplify this by using the fact that ln(|cos(x)|) = ln(cos(x)) if x is in the range (0, π/2) or (3π/2, 2π). However, if x is in the range (π/2, 3π/2), then ln(|cos(x)|) = ln(-cos(x)) = ln(cos(x)) – ln(-1) = ln(cos(x)) + iπ.

Therefore, the final result is:

∫sin(x)dx = -cos(x) + C, for x in (0, π) or (π, 2π)

or

∫sin(x)dx = cos(x) + C, for x in (π/2, 3π/2)

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