What is the integral of ln(cos(x)) ?

The integral of ln(cos(x)) is 1/2 i\Li_{2}(-e^{2ix})+(ix^2)/2-xln(1+(-1)^{(2x)/pi})+xln(cos(x))+C

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integral of ln(cos(x))

\int ln (cos (x)) d x

\frac{1}{2} i Li_{2} (- e^{2 i x}) + \frac{i x ^{2}}{2} - x ln (1 + (- 1)^{\frac{2 x}{π}}) + x ln (cos (x)) + C

Solution steps

\int ln (cos (x)) d x

Use the nonelementary integral:

\int ln (cos (x)) d x = \frac{1}{2} i Li_{2} (- e^{2 i x}) + \frac{i x ^{2}}{2} - x ln (1 + (- 1)^{\frac{2 x}{π}}) + x ln (cos (x))

= \frac{1}{2} i Li_{2} (- e^{2 i x}) + \frac{i x ^{2}}{2} - x ln (1 + (- 1)^{\frac{2 x}{π}}) + x ln (cos (x))

Add a constant to the solution

= \frac{1}{2} i Li_{2} (- e^{2 i x}) + \frac{i x ^{2}}{2} - x ln (1 + (- 1)^{\frac{2 x}{π}}) + x ln (cos (x)) + C

y^{^{'}} = 1 - \frac{y}{100}

sin (x)

5 (x^{2 n - 1}) \cdot (2 x^{3 n + 1})^{2}

f (x) = x e^{\frac{1}{x}}

x \to 0 lim (\frac{e ^{- 1 x} - 1}{x})

What is the integral of ln(cos(x)) ?
The integral of ln(cos(x)) is 1/2 i\Li_{2}(-e^{2ix})+(ix^2)/2-xln(1+(-1)^{(2x)/pi})+xln(cos(x))+C