integral of (y^2+34y+289)/((y^2+289)^2)

解答

\int \frac{y ^{2} + 34 y + 289}{( y ^{2} + 289 ) ^{2}} d y

\frac{1}{17} arctan (\frac{y}{17}) - \frac{17}{y ^{2} + 289} + C

求解步骤

\int \frac{y ^{2} + 34 y + 289}{( y ^{2} + 289 ) ^{2}} d y

将

\frac{y ^{2} + 34 y + 289}{( y ^{2} + 289 ) ^{2}}

用部份分式展开:

\frac{1}{y ^{2} + 289} + \frac{34 y}{( y ^{2} + 289 ) ^{2}}

= \int \frac{1}{y ^{2} + 289} + \frac{34 y}{( y ^{2} + 289 ) ^{2}} 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{1}{y ^{2} + 289} d y + \int \frac{34 y}{( y ^{2} + 289 ) ^{2}} d y

\int \frac{1}{y ^{2} + 289} d y = \frac{1}{17} arctan (\frac{y}{17})

\int \frac{34 y}{( y ^{2} + 289 ) ^{2}} d y = - \frac{17}{y ^{2} + 289}

= \frac{1}{17} arctan (\frac{y}{17}) - \frac{17}{y ^{2} + 289}

解答补常数

= \frac{1}{17} arctan (\frac{y}{17}) - \frac{17}{y ^{2} + 289} + C

d er i v a t i v e (csc (x) + cot (x))^{- 1}

x \to \infty lim (6)

\frac{d}{d t} (\frac{t + 1}{t - 1})

a re a x, x^{\frac{1}{2}}, 1, 2

\int 2 5 x^{3} - 3 x d x