integral of 1/(cos^2(5x))

解答

\int \frac{1}{cos ^{2} ( 5 x )} d x

\frac{1}{5} tan (5 x) + C

求解步骤

\int \frac{1}{cos ^{2} ( 5 x )} d x

使用三角恒等式改写

= \int sec^{2} (5 x) d x

使用换元积分法

= \int sec^{2} (u) \frac{1}{5} d u

提出常数:

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

= \frac{1}{5} \cdot \int sec^{2} (u) d u

使用积分常见定则:

\int sec^{2} (u) d u = tan (u)

= \frac{1}{5} tan (u)

u = 5 x

代回

= \frac{1}{5} tan (5 x)

解答补常数

= \frac{1}{5} tan (5 x) + C

x \to 1 lim (\frac{2 \cdot x + 3}{x + 4})

\int \frac{x + 1}{3 - 2 x - x ^{2}} d x

\frac{\partial}{\partial y} (e^{x y} tan (y))

\frac{d}{d x} (1 + \frac{5 e ^{- 2 x^{2}}}{2 - ∣ x ∣})

d er i v a t i v e y = cos (sin (tan (7 x)))