integral of (y^2+26y+169)/((y^2+169)^2)

解答

\int \frac{y ^{2} + 26 y + 169}{( y ^{2} + 169 ) ^{2}} d y

\frac{1}{13} arctan (\frac{y}{13}) - \frac{13}{y ^{2} + 169} + C

求解步骤

\int \frac{y ^{2} + 26 y + 169}{( y ^{2} + 169 ) ^{2}} d y

将

\frac{y ^{2} + 26 y + 169}{( y ^{2} + 169 ) ^{2}}

用部份分式展开:

\frac{1}{y ^{2} + 169} + \frac{26 y}{( y ^{2} + 169 ) ^{2}}

= \int \frac{1}{y ^{2} + 169} + \frac{26 y}{( y ^{2} + 169 ) ^{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} + 169} d y + \int \frac{26 y}{( y ^{2} + 169 ) ^{2}} d y

\int \frac{1}{y ^{2} + 169} d y = \frac{1}{13} arctan (\frac{y}{13})

\int \frac{26 y}{( y ^{2} + 169 ) ^{2}} d y = - \frac{13}{y ^{2} + 169}

= \frac{1}{13} arctan (\frac{y}{13}) - \frac{13}{y ^{2} + 169}

解答补常数

= \frac{1}{13} arctan (\frac{y}{13}) - \frac{13}{y ^{2} + 169} + C

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