integral of 1/((1+tan(x)))

解答

\int \frac{1}{( 1 + tan ( x ) )} d x

\frac{1}{2} (- \frac{1}{2} ln (sec^{2} (x)) + arctan (tan (x))) + \frac{1}{2} ln ∣ tan (x) + 1 ∣ + C

求解步骤

\int \frac{1}{1 + tan ( x )} d x

使用换元积分法

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

将

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

用部份分式展开:

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

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

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

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

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

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

= \frac{1}{2} (- \frac{1}{2} ln u^{2} + 1 + arctan (u)) + \frac{1}{2} ln ∣ u + 1 ∣

u = tan (x)

代回

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

\frac{1}{2} ln tan^{2} (x) + 1 = \frac{1}{2} ln (sec^{2} (x))

= \frac{1}{2} (- \frac{1}{2} ln (sec^{2} (x)) + arctan (tan (x))) + \frac{1}{2} ln ∣ tan (x) + 1 ∣

解答补常数

= \frac{1}{2} (- \frac{1}{2} ln (sec^{2} (x)) + arctan (tan (x))) + \frac{1}{2} ln ∣ tan (x) + 1 ∣ + C

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