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受欢迎的微积分 >

integral of (sin(2x)*sinh(2x))/2

解答

\int \frac{sin ( 2 x ) \cdot sinh ( 2 x )}{2} d x

解答

\frac{1}{8} (- cos (2 x) sinh (2 x) + sin (2 x) cosh (2 x)) + C

求解步骤

\int \frac{sin ( 2 x ) sinh ( 2 x )}{2} d x

提出常数:

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

= \frac{1}{2} \cdot \int sin (2 x) sinh (2 x) d x

使用分布积分法

= \frac{1}{2} (- \frac{1}{2} sinh (2 x) cos (2 x) - \int - cosh (2 x) cos (2 x) d x)

提出常数:

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

= \frac{1}{2} (- \frac{1}{2} sinh (2 x) cos (2 x) - (- \int cosh (2 x) cos (2 x) d x))

使用分布积分法

= \frac{1}{2} (- \frac{1}{2} sinh (2 x) cos (2 x) - (- (\frac{1}{2} cosh (2 x) sin (2 x) - \int sinh (2 x) sin (2 x) d x)))

因此

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

整理

\int sinh (2 x) sin (2 x) d x

= \frac{1}{2} (- \frac{cos ( 2 x ) sinh ( 2 x )}{4} + \frac{sin ( 2 x ) cosh ( 2 x )}{4})

化简

\frac{1}{2} (- \frac{cos ( 2 x ) sinh ( 2 x )}{4} + \frac{sin ( 2 x ) cosh ( 2 x )}{4}) : \frac{1}{8} (- cos (2 x) sinh (2 x) + sin (2 x) cosh (2 x))

= \frac{1}{8} (- cos (2 x) sinh (2 x) + sin (2 x) cosh (2 x))

解答补常数

= \frac{1}{8} (- cos (2 x) sinh (2 x) + sin (2 x) cosh (2 x)) + C

作图

流行的例子

integral of cos(x)(7+4sin^2(x))

\int cos (x) (7 + 4 sin^{2} (x)) d x

integral of 2sin(4x)

\int 2 sin (4 x) d x

derivative f(x)=sqrt(x-4)

d er i v a t i v e f (x) = x - 4

(2xy^2+2y)dx+(2x^2y+2x)dy=0

(2 x y^{2} + 2 y) d x + (2 x^{2} y + 2 x) d y = 0

limit as x approaches 4 of ((3x^2-17x+20))/(4x^2-25x+36)

x \to 4 lim (\frac{( 3 x ^{2} - 17 x + 20 )}{4 x ^{2} - 25 x + 36})