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

The integral of ln(cos(t)) is 1/2 i\Li_{2}(-e^{2it})+(it^2)/2-tln(1+(-1)^{(2t)/pi})+tln(cos(t))+C

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integral of ln(cos(t))

\int ln (cos (t)) d t

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

Solution steps

\int ln (cos (t)) d t

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 t}) + \frac{i t ^{2}}{2} - t ln (1 + (- 1)^{\frac{2 t}{π}}) + t ln (cos (t))

Add a constant to the solution

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

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