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

The integral of-ln(cos(x)) is -i 1/2 \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

- i \frac{1}{2} 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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \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)))

Simplify

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

= - i \frac{1}{2} 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

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

x \to 0 lim (2 x - x ln (x))

\frac{d}{d x} (3 x^{2} + 2 x + 5)

x \to 0 lim ((7 x)^{3 x})

\int - cos (x) + sin (x) + 3 d x

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