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受欢迎的微积分 >

integral from 8 to 14 of y/(y^2-4y-5)

解答

\int_{8}^{14} \frac{y}{y ^{2} - 4 y - 5} d y

解答

\frac{ln ( \frac{5}{3} ) + 5 ln ( 3 )}{6}

+1

十进制

1.00064 \dots

求解步骤

\int_{8}^{14} \frac{y}{y ^{2} - 4 y - 5} d y

将

\frac{y}{y ^{2} - 4 y - 5}

用部份分式展开:

\frac{1}{6 ( y + 1 )} + \frac{5}{6 ( y - 5 )}

= \int_{8}^{14} \frac{1}{6 ( y + 1 )} + \frac{5}{6 ( y - 5 )} d y

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{8}^{14} \frac{1}{6 ( y + 1 )} d y + \int_{8}^{14} \frac{5}{6 ( y - 5 )} d y

\int_{8}^{14} \frac{1}{6 ( y + 1 )} d y = \frac{1}{6} ln (\frac{5}{3})

\int_{8}^{14} \frac{5}{6 ( y - 5 )} d y = \frac{5}{6} ln (3)

= \frac{1}{6} ln (\frac{5}{3}) + \frac{5}{6} ln (3)

化简

\frac{1}{6} ln (\frac{5}{3}) + \frac{5}{6} ln (3) : \frac{ln ( \frac{5}{3} ) + 5 ln ( 3 )}{6}

= \frac{ln ( \frac{5}{3} ) + 5 ln ( 3 )}{6}

作图

流行的例子

y^{^{'}} = e^{- 2 x}

tangent f(x)=x^2+1

t an g e n t f (x) = x^{2} + 1

7y^{''}+5y^'+7y=0

7 y^{^{''}} + 5 y^{^{'}} + 7 y = 0

limit as h approaches 0 of (4/(2+h)-2)/h

h \to 0 lim (\frac{\frac{4}{2 + h} - 2}{h})

面积 x=1-y^2,-x-1

a re a x = 1 - y^{2}, - x - 1