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受欢迎的微积分 >

integral of 2/(35sin(x)-12cos(x))

解答

\int \frac{2}{35 sin ( x ) - 12 cos ( x )} d x

解答

- \frac{2}{37} (ln \frac{12}{37} tan (\frac{x}{2}) + \frac{72}{37} - ln \frac{12}{37} tan (\frac{x}{2}) - \frac{2}{37}) + C

求解步骤

\int \frac{2}{35 sin ( x ) - 12 cos ( x )} d x

提出常数:

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

= 2 \cdot \int \frac{1}{35 sin ( x ) - 12 cos ( x )} d x

使用换元积分法

= 2 \cdot \int \frac{1}{35 u - 6 ( - u ^{2} + 1 )} d u

对

35 u - 6 (- u^{2} + 1)

配方:

6 (u + \frac{35}{12})^{2} - \frac{1369}{24}

= 2 \cdot \int \frac{1}{6 ( u + \frac{35}{12} ) ^{2} - \frac{1369}{24}} d u

使用换元积分法

= 2 \cdot \int \frac{24}{144 v ^{2} - 1369} d v

提出常数:

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

= 2 \cdot 24 \cdot \int \frac{1}{144 v ^{2} - 1369} d v

使用换元积分法

= 2 \cdot 24 \cdot \int \frac{1}{444 ( w ^{2} - 1 )} d w

提出常数:

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

= 2 \cdot 24 \cdot \frac{1}{444} \cdot \int \frac{1}{w ^{2} - 1} d w

因式分解

w^{2} - 1 : - (- w^{2} + 1)

= 2 \cdot 24 \cdot \frac{1}{444} \cdot \int \frac{1}{- ( - w ^{2} + 1 )} d w

提出常数:

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

= 2 \cdot 24 \cdot \frac{1}{444} (- \int \frac{1}{- w ^{2} + 1} d w)

使用积分常见定则:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= 2 \cdot 24 \cdot \frac{1}{444} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

代回

= 2 \cdot 24 \cdot \frac{1}{444} (- (\frac{ln \frac{12}{37} ( tan ( \frac{x}{2} ) + \frac{35}{12} ) + 1}{2} - \frac{ln \frac{12}{37} ( tan ( \frac{x}{2} ) + \frac{35}{12} ) - 1}{2}))

化简

2 \cdot 24 \cdot \frac{1}{444} (- (\frac{ln \frac{12}{37} ( tan ( \frac{x}{2} ) + \frac{35}{12} ) + 1}{2} - \frac{ln \frac{12}{37} ( tan ( \frac{x}{2} ) + \frac{35}{12} ) - 1}{2})) : - \frac{2}{37} (ln \frac{12}{37} tan (\frac{x}{2}) + \frac{72}{37} - ln \frac{12}{37} tan (\frac{x}{2}) - \frac{2}{37})

= - \frac{2}{37} (ln \frac{12}{37} tan (\frac{x}{2}) + \frac{72}{37} - ln \frac{12}{37} tan (\frac{x}{2}) - \frac{2}{37})

解答补常数

= - \frac{2}{37} (ln \frac{12}{37} tan (\frac{x}{2}) + \frac{72}{37} - ln \frac{12}{37} tan (\frac{x}{2}) - \frac{2}{37}) + C

作图

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