integral of sec(x/2)

解答

\int sec (\frac{x}{2}) d x

2 ln tan (\frac{x}{2}) + sec (\frac{x}{2}) + C

求解步骤

\int sec (\frac{x}{2}) d x

使用三角恒等式改写

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

使用换元积分法

= \int 2 sec (u) d u

提出常数:

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

= 2 \cdot \int sec (u) d u

使用积分常见定则:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= 2 ln ∣ tan (u) + sec (u) ∣

u = \frac{x}{2}

代回

= 2 ln tan (\frac{x}{2}) + sec (\frac{x}{2})

解答补常数

= 2 ln tan (\frac{x}{2}) + sec (\frac{x}{2}) + C

d er i v a t i v e 7 e^{x} + \frac{4}{3 x}

t an g e n t f (x) = 110 x - 16 x^{2}, at 3.438

d er i v a t i v e \frac{x ^{4} - 16 x ^{2} - 16}{( x ^{2} - 4 ) ^{2}}

x y d y - (x^{2} + y^{2}) d x = 0

n = 1 \sum \infty \frac{1}{( 0.8 ) ^{n}}