integral of (x+5)/(sqrt(x^2+9))

解答

\int \frac{x + 5}{x ^{2} + 9} d x

x^{2} + 9 + 5 ln (\frac{1}{3} x + 9 + x^{2}) + C

求解步骤

\int \frac{x + 5}{x ^{2} + 9} d x

乘开

\frac{x + 5}{x ^{2} + 9} : \frac{x}{x ^{2} + 9} + \frac{5}{x ^{2} + 9}

= \int \frac{x}{x ^{2} + 9} + \frac{5}{x ^{2} + 9} d x

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{x}{x ^{2} + 9} d x + \int \frac{5}{x ^{2} + 9} d x

\int \frac{x}{x ^{2} + 9} d x = x^{2} + 9

\int \frac{5}{x ^{2} + 9} d x = 5 ln (\frac{1}{3} x + 9 + x^{2})

= x^{2} + 9 + 5 ln (\frac{1}{3} x + 9 + x^{2})

解答补常数

= x^{2} + 9 + 5 ln (\frac{1}{3} x + 9 + x^{2}) + C

\int \frac{x - 1}{( x - 2 ) ( x - 3 )} d x

y^{^{'}} + e^{x} y = 7 e^{x}

d er i v a t i v e (7 x + 1) (x^{2} - 6)

\frac{d}{d y} (a^{2} - y^{2})

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