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受欢迎的微积分 >

integral of (cos(y))/(sin^2(y)+sin(y)-2)

解答

\int \frac{cos ( y )}{sin ^{2} ( y ) + sin ( y ) - 2} d y

解答

- \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} sin (y) - ln - \frac{2}{3} + \frac{2}{3} sin (y)) + C

求解步骤

\int \frac{cos ( y )}{sin ^{2} ( y ) + sin ( y ) - 2} d y

使用换元积分法

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

对

u^{2} + u - 2

配方:

(u + \frac{1}{2})^{2} - \frac{9}{4}

= \int \frac{1}{( u + \frac{1}{2} ) ^{2} - \frac{9}{4}} d u

使用换元积分法

= \int \frac{4}{4 v ^{2} - 9} d v

提出常数:

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

= 4 \cdot \int \frac{1}{4 v ^{2} - 9} d v

使用换元积分法

= 4 \cdot \int \frac{1}{6 ( 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 \frac{1}{6} \cdot \int \frac{1}{w ^{2} - 1} d w

因式分解

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

= 4 \cdot \frac{1}{6} \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 \frac{1}{6} (- \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 \frac{1}{6} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

代回

= 4 \cdot \frac{1}{6} (- (\frac{ln \frac{2}{3} ( sin ( y ) + \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{3} ( sin ( y ) + \frac{1}{2} ) - 1}{2}))

化简

4 \cdot \frac{1}{6} (- (\frac{ln \frac{2}{3} ( sin ( y ) + \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{3} ( sin ( y ) + \frac{1}{2} ) - 1}{2})) : - \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} sin (y) - ln - \frac{2}{3} + \frac{2}{3} sin (y))

= - \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} sin (y) - ln - \frac{2}{3} + \frac{2}{3} sin (y))

解答补常数

= - \frac{1}{3} (ln \frac{4}{3} + \frac{2}{3} sin (y) - ln - \frac{2}{3} + \frac{2}{3} sin (y)) + C

作图

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