integral of (10/9 x+14)/(x^2+9)

解答

\int \frac{\frac{10}{9} x + 14}{x ^{2} + 9} d x

\frac{5}{9} ln x^{2} + 9 + \frac{14}{3} arctan (\frac{x}{3}) + C

求解步骤

\int \frac{\frac{10}{9} x + 14}{x ^{2} + 9} d x

乘开

\frac{\frac{10}{9} x + 14}{x ^{2} + 9} : \frac{\frac{10}{9} x}{x ^{2} + 9} + \frac{14}{x ^{2} + 9}

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{\frac{10}{9} x}{x ^{2} + 9} d x + \int \frac{14}{x ^{2} + 9} d x

\int \frac{\frac{10}{9} x}{x ^{2} + 9} d x = \frac{5}{9} ln x^{2} + 9

\int \frac{14}{x ^{2} + 9} d x = \frac{14}{3} arctan (\frac{x}{3})

= \frac{5}{9} ln x^{2} + 9 + \frac{14}{3} arctan (\frac{x}{3})

解答补常数

= \frac{5}{9} ln x^{2} + 9 + \frac{14}{3} arctan (\frac{x}{3}) + C

\int \frac{sinh ( ln ( x ) )}{x} d x

x \to + 0 lim (\frac{∣ x - 3 ∣}{x - 3})

\int \frac{y ^{2} + 28 y + 196}{( y ^{2} + 196 ) ^{2}} d y

x \to - 4 lim (\frac{x ^{2} - 1}{4 - x})

\int_{0}^{1} cos (\frac{π x}{2}) d x