integral of t/(2-t)

解答

\int \frac{t}{2 - t} d t

- 2 ln ∣ 2 - t ∣ - t + C

求解步骤

\int \frac{t}{2 - t} d t

\frac{t}{2 - t} = \frac{2}{2 - t} - 1

= \int \frac{2}{2 - t} - 1 d t

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}{2 - t} d t - \int 1 d t

\int \frac{2}{2 - t} d t = - 2 ln ∣ 2 - t ∣

\int 1 d t = t

= - 2 ln ∣ 2 - t ∣ - t

解答补常数

= - 2 ln ∣ 2 - t ∣ - t + C

\int \frac{1 - x ^{4}}{x ^{5}} d x

\int \frac{6 x ^{4} + x - 3}{2 x ^{3} + 2 x ^{2} - 12 x} d x

\int \frac{7 x ^{2}}{( x ^{2} + 3 ) ^{2}} d x

\int \frac{10}{x ^{3} - 4 x ^{2} + 8 x} d x

\int (x + 2) 6 - x d x