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

解答

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

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

求解步骤

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

使用换元积分法

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

提出常数:

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

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

乘开

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

= \frac{1}{2} \cdot \int \frac{1}{u} + u 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

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

\int \frac{1}{u} d u = ln ∣ u ∣

\int u d u = \frac{u ^{2}}{2}

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

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

代回

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

解答补常数

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

\int (x + 3) (x + 1)^{\frac{1}{2}} d x

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\int x (x^{2} - 1)^{4} d x

\int \frac{3}{9 x ^{2} + 1} d x