integral of 1/(cos^2(t)sqrt(1+tan(t)))

解答

\int \frac{1}{cos ^{2} ( t ) 1 + tan ( t )} d t

2 1 + tan (t) + C

求解步骤

\int \frac{1}{cos ^{2} ( t ) 1 + tan ( t )} d t

使用三角恒等式改写

= \int \frac{sec ^{2} ( t )}{1 + tan ( t )} d t

使用换元积分法

= \int 2 d u

常数积分:

\int a d x = a x

= 2 u

u = 1 + tan (t)

代回

= 2 1 + tan (t)

解答补常数

= 2 1 + tan (t) + C