integral of ((y^2+1))/(y^2+4)

解答

\int \frac{( y ^{2} + 1 )}{y ^{2} + 4} d y

- \frac{3}{2} arctan (\frac{y}{2}) + y + C

求解步骤

\int \frac{y ^{2} + 1}{y ^{2} + 4} d y

\frac{y ^{2} + 1}{y ^{2} + 4} = - \frac{3}{y ^{2} + 4} + 1

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

\int \frac{3}{y ^{2} + 4} d y = \frac{3}{2} arctan (\frac{y}{2})

\int 1 d y = y

= - \frac{3}{2} arctan (\frac{y}{2}) + y

解答补常数

= - \frac{3}{2} arctan (\frac{y}{2}) + y + C

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

\frac{d}{d x} (x^{2} y^{4})

x \to 2 lim (x^{2} - 5)

x \to 0 lim (\frac{1 - e ^{- x}}{x})

\frac{d}{d x} (\frac{x ^{2}}{x ^{3}})