integral of-ln(cos(x))

解答

\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

求解步骤

\int - ln (cos (x)) d x

提出常数:

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

= - \int ln (cos (x)) d x

这是非初等函数积分:

\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)))

化简

- (\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))

解答补常数

= - 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))

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

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

\int \frac{cos ( x )}{sin ^{6} ( x )} d x

\frac{d y}{d x} = \frac{x}{18 y}