integral of ((2+w))/(sqrt(13+12w-w^2))

解答

\int \frac{( 2 + w )}{13 + 12 w - w ^{2}} d w

8 arcsin (\frac{1}{7} (w - 6)) - - w^{2} + 12 w + 13 + C

求解步骤

\int \frac{2 + w}{13 + 12 w - w ^{2}} d w

乘开

\frac{2 + w}{13 + 12 w - w ^{2}} : \frac{2}{13 + 12 w - w ^{2}} + \frac{w}{13 + 12 w - w ^{2}}

= \int \frac{2}{13 + 12 w - w ^{2}} + \frac{w}{13 + 12 w - w ^{2}} d w

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{2}{13 + 12 w - w ^{2}} d w + \int \frac{w}{13 + 12 w - w ^{2}} d w

\int \frac{2}{13 + 12 w - w ^{2}} d w = 2 arcsin (\frac{1}{7} (w - 6))

\int \frac{w}{13 + 12 w - w ^{2}} d w = - - w^{2} + 12 w + 13 + 6 arcsin (\frac{1}{7} (w - 6))

= 2 arcsin (\frac{1}{7} (w - 6)) - - w^{2} + 12 w + 13 + 6 arcsin (\frac{1}{7} (w - 6))

化简

= 8 arcsin (\frac{1}{7} (w - 6)) - - w^{2} + 12 w + 13

解答补常数

= 8 arcsin (\frac{1}{7} (w - 6)) - - w^{2} + 12 w + 13 + C

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