# What's the integral of #int tanx / (secx + cosx)dx#?

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To solve the integral ∫tan(x) / (sec(x) + cos(x)) dx, perform the following steps:

- Rewrite tan(x) as sin(x) / cos(x).
- Rewrite sec(x) as 1 / cos(x).
- Rewrite the integral as ∫(sin(x) / cos(x)) / (1 / cos(x) + cos(x)) dx.
- Simplify the denominator to obtain ∫(sin(x) / cos(x)) / (1 + cos^2(x)) dx.
- Multiply the numerator and denominator by cos(x) to rationalize the expression: ∫sin(x) / (cos(x) + cos^3(x)) dx.
- Perform a substitution, letting u = cos(x), du = -sin(x) dx.
- Rewrite the integral in terms of u: ∫-du / (u + u^3) = -∫du / (u(1 + u^2)).
- Perform partial fraction decomposition on the denominator to obtain: ∫(-1/u) / (1 + u^2) du - ∫(1/(1 + u^2)) du.
- Integrate each term: -ln|u| - arctan(u) + C.
- Substitute back u = cos(x): -ln|cos(x)| - arctan(cos(x)) + C.

So, the integral of tan(x) / (sec(x) + cos(x)) dx is -ln|cos(x)| - arctan(cos(x)) + C, where C is the constant of integration.

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