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受欢迎的微积分 >

integral of sqrt(1+csc(x))

解答

\int 1 + csc (x) d x

解答

2 (2 arctan (2 tan (\frac{x}{2}) + 1) + 2 arctan (2 tan (\frac{x}{2}) - 1)) + C

求解步骤

\int 1 + csc (x) d x

使用三角恒等式改写

= \int 1 + \frac{1}{sin ( x )} d x

使用换元积分法

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

提出常数:

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

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

乘开

\frac{u + 1}{u ( 1 + u ^{2} )} : \frac{u}{u ( 1 + u ^{2} )} + \frac{1}{u ( 1 + u ^{2} )}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{u}{u ( 1 + u ^{2} )} d u = 22 (- \frac{1}{8} (ln 2 u + 22 u + 2 - 2 arctan (2 u + 1)) + \frac{1}{8} (ln 2 u - 22 u + 2 + 2 arctan (2 u - 1)))

\int \frac{1}{u ( 1 + u ^{2} )} d u = 2 (\frac{1}{4 2} (ln 2 u + 22 u + 2 + 2 arctan (2 u + 1)) + \frac{1}{4 2} (- ln 2 u - 22 u + 2 + 2 arctan (2 u - 1)))

= 2 (22 (- \frac{1}{8} (ln 2 u + 22 u + 2 - 2 arctan (2 u + 1)) + \frac{1}{8} (ln 2 u - 22 u + 2 + 2 arctan (2 u - 1))) + 2 (\frac{1}{4 2} (ln 2 u + 22 u + 2 + 2 arctan (2 u + 1)) + \frac{1}{4 2} (- ln 2 u - 22 u + 2 + 2 arctan (2 u - 1))))

u = tan (\frac{x}{2})

代回

= 2 (22 (- \frac{1}{8} (ln 2 tan (\frac{x}{2}) + 22 tan (\frac{x}{2}) + 2 - 2 arctan (2 tan (\frac{x}{2}) + 1)) + \frac{1}{8} (ln 2 tan (\frac{x}{2}) - 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) - 1))) + 2 (\frac{1}{4 2} (ln 2 tan (\frac{x}{2}) + 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) + 1)) + \frac{1}{4 2} (- ln 2 tan (\frac{x}{2}) - 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) - 1))))

乘开

22 (- \frac{1}{8} (ln 2 tan (\frac{x}{2}) + 22 tan (\frac{x}{2}) + 2 - 2 arctan (2 tan (\frac{x}{2}) + 1)) + \frac{1}{8} (ln 2 tan (\frac{x}{2}) - 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) - 1))) + 2 (\frac{1}{4 2} (ln 2 tan (\frac{x}{2}) + 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) + 1)) + \frac{1}{4 2} (- ln 2 tan (\frac{x}{2}) - 22 tan (\frac{x}{2}) + 2 + 2 arctan (2 tan (\frac{x}{2}) - 1))) : 2 arctan (2 tan (\frac{x}{2}) + 1) + 2 arctan (2 tan (\frac{x}{2}) - 1)

= 2 (2 arctan (2 tan (\frac{x}{2}) + 1) + 2 arctan (2 tan (\frac{x}{2}) - 1))

解答补常数

= 2 (2 arctan (2 tan (\frac{x}{2}) + 1) + 2 arctan (2 tan (\frac{x}{2}) - 1)) + C

作图

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