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受欢迎的微积分 >

integral of (cos(x))/(3sin^2(x)+4sin(x))

解答

\int \frac{cos ( x )}{3 sin ^{2} ( x ) + 4 sin ( x )} d x

解答

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

求解步骤

\int \frac{cos ( x )}{3 sin ^{2} ( x ) + 4 sin ( x )} d x

使用换元积分法

= \int \frac{1}{3 u ^{2} + 4 u} d u

对

3 u^{2} + 4 u

配方:

3 (u + \frac{2}{3})^{2} - \frac{4}{3}

= \int \frac{1}{3 ( u + \frac{2}{3} ) ^{2} - \frac{4}{3}} d u

使用换元积分法

= \int \frac{3}{9 v ^{2} - 4} d v

提出常数:

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

= 3 \cdot \int \frac{1}{9 v ^{2} - 4} d v

使用换元积分法

= 3 \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

= 3 \cdot \frac{1}{6} \cdot \int \frac{1}{w ^{2} - 1} d w

因式分解

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

= 3 \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

= 3 \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}

= 3 \cdot \frac{1}{6} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

integral of 3x^2+4x+8

\int 3 x^{2} + 4 x + 8 d x

integral of (64)/((x^2+4)^2)

\int \frac{64}{( x ^{2} + 4 ) ^{2}} d x

integral of 5/((t^2-25)^2)

\int \frac{5}{( t ^{2} - 25 ) ^{2}} d t

integral of (20x)/((x^2+1)^2)

\int \frac{20 x}{( x ^{2} + 1 ) ^{2}} d x

integral of (2x)/((2x-1)^2)

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