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受欢迎的微积分 >

integral of 4/(15sin(x)-8cos(x))

解答

\int \frac{4}{15 sin ( x ) - 8 cos ( x )} d x

解答

- \frac{4}{17} (ln \frac{8}{17} tan (\frac{x}{2}) + \frac{32}{17} - ln \frac{8}{17} tan (\frac{x}{2}) - \frac{2}{17}) + C

求解步骤

\int \frac{4}{15 sin ( x ) - 8 cos ( x )} d x

提出常数:

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

= 4 \cdot \int \frac{1}{15 sin ( x ) - 8 cos ( x )} d x

使用换元积分法

= 4 \cdot \int \frac{1}{15 u - 4 ( - u ^{2} + 1 )} d u

对

15 u - 4 (- u^{2} + 1)

配方:

4 (u + \frac{15}{8})^{2} - \frac{289}{16}

= 4 \cdot \int \frac{1}{4 ( u + \frac{15}{8} ) ^{2} - \frac{289}{16}} d u

使用换元积分法

= 4 \cdot \int \frac{16}{64 v ^{2} - 289} d v

提出常数:

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

= 4 \cdot 16 \cdot \int \frac{1}{64 v ^{2} - 289} d v

使用换元积分法

= 4 \cdot 16 \cdot \int \frac{1}{136 ( w ^{2} - 1 )} d w

提出常数:

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

= 4 \cdot 16 \cdot \frac{1}{136} \cdot \int \frac{1}{w ^{2} - 1} d w

因式分解

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

= 4 \cdot 16 \cdot \frac{1}{136} \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

= 4 \cdot 16 \cdot \frac{1}{136} (- \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}

= 4 \cdot 16 \cdot \frac{1}{136} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

代回

= 4 \cdot 16 \cdot \frac{1}{136} (- (\frac{ln \frac{8}{17} ( tan ( \frac{x}{2} ) + \frac{15}{8} ) + 1}{2} - \frac{ln \frac{8}{17} ( tan ( \frac{x}{2} ) + \frac{15}{8} ) - 1}{2}))

化简

4 \cdot 16 \cdot \frac{1}{136} (- (\frac{ln \frac{8}{17} ( tan ( \frac{x}{2} ) + \frac{15}{8} ) + 1}{2} - \frac{ln \frac{8}{17} ( tan ( \frac{x}{2} ) + \frac{15}{8} ) - 1}{2})) : - \frac{4}{17} (ln \frac{8}{17} tan (\frac{x}{2}) + \frac{32}{17} - ln \frac{8}{17} tan (\frac{x}{2}) - \frac{2}{17})

= - \frac{4}{17} (ln \frac{8}{17} tan (\frac{x}{2}) + \frac{32}{17} - ln \frac{8}{17} tan (\frac{x}{2}) - \frac{2}{17})

解答补常数

= - \frac{4}{17} (ln \frac{8}{17} tan (\frac{x}{2}) + \frac{32}{17} - ln \frac{8}{17} tan (\frac{x}{2}) - \frac{2}{17}) + C

作图

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