integral of sqrt(cos(2t))

解答

\int cos (2 t) d t

E (2∣ \frac{2 t}{2}) + C

求解步骤

\int cos (2 t) d t

使用换元积分法

= \int cos (u) \frac{1}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int cos (u) d u

这是非初等函数积分:

\int cos (x) d x = 2 E (\frac{x}{2} ∣ 2)

= \frac{1}{2} \cdot 2 E (\frac{u}{2} ∣ 2)

u = 2 t

代回

= \frac{1}{2} \cdot 2 E (\frac{2 t}{2} ∣ 2)

化简

\frac{1}{2} \cdot 2 E (\frac{2 t}{2} ∣ 2) : E (2∣ \frac{2 t}{2})

= E (2∣ \frac{2 t}{2})

解答补常数

= E (2∣ \frac{2 t}{2}) + C

\int y^{- \frac{3}{2}} d y

\int - 2 t + 3 d t

\int ln (6) d x

\int \frac{1 + 3 x}{x} d x

\int - 2 e^{\frac{t}{4}} d t