integral of ln(s)ecxtan(x)

解答

\int ln (s) ec x tan (x) d x

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

求解步骤

\int ln (s) ec x tan (x) d x

提出常数:

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

= ln (s) ce \cdot \int tan (x) x d x

这是非初等函数积分:

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

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

使用虚数运算法则:

e^{ia} = cos (a) + i sin (a)

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

解答补常数

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

t an g e n t 10 x^{3} - 30 x + 120

\frac{d}{d x} (2 x^{2} y^{2})

d er i v a t i v e z + 3 (\frac{1}{z ^{2}})

\frac{d y}{d x} + y tan (x) = 2 x cos (x)

\frac{\partial}{\partial z} (x y + yz + x z)