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受欢迎的微积分 >

integral of (sin(x))/(sin(x)+cos(x)+1)

解答

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

解答

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

求解步骤

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

使用换元积分法

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

将

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

用部份分式展开:

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

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

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

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

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

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

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

代回

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

化简

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

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

解答补常数

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

作图

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