integral of ((ax+b))/((cx+)

解答

\int \frac{( a x + b )}{( c x + d )} d x

a \frac{1}{c ^{2}} (c x + d - d ln ∣ c x + d ∣) + b \frac{1}{c} ln ∣ c x + d ∣ + C

求解步骤

\int \frac{a x + b}{c x + d} d x

乘开

\frac{a x + b}{c x + d} : \frac{a x}{c x + d} + \frac{b}{c x + d}

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{a x}{c x + d} d x + \int \frac{b}{c x + d} d x

\int \frac{a x}{c x + d} d x = a \frac{1}{c ^{2}} (c x + d - d ln ∣ c x + d ∣)

\int \frac{b}{c x + d} d x = b \frac{1}{c} ln ∣ c x + d ∣

= a \frac{1}{c ^{2}} (c x + d - d ln ∣ c x + d ∣) + b \frac{1}{c} ln ∣ c x + d ∣

解答补常数

= a \frac{1}{c ^{2}} (c x + d - d ln ∣ c x + d ∣) + b \frac{1}{c} ln ∣ c x + d ∣ + C

d er i v a t i v e \frac{7}{2 x ^{2}}

x \to \infty lim (arctan (x^{6} - x^{9}))

d er i v a t i v e y = arctan (3 x)

t an g e n t f (x) = \frac{3 + 4 x ^{\frac{1}{2}}}{2 + x ^{\frac{3}{2}}}

\int \frac{x ^{2}}{( 5 - x ^{2} ) ^{3}} d x