integral of 1/(12+4x-x^2)

解答

\int \frac{1}{12 + 4 x - x ^{2}} d x

\frac{1}{8} ln ∣ x + 2 ∣ - \frac{1}{8} ln ∣ x - 6 ∣ + C

求解步骤

\int \frac{1}{12 + 4 x - x ^{2}} d x

将

\frac{1}{12 + 4 x - x ^{2}}

用部份分式展开:

\frac{1}{8 ( x + 2 )} - \frac{1}{8 ( x - 6 )}

= \int \frac{1}{8 ( x + 2 )} - \frac{1}{8 ( x - 6 )} 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{1}{8 ( x + 2 )} d x - \int \frac{1}{8 ( x - 6 )} d x

\int \frac{1}{8 ( x + 2 )} d x = \frac{1}{8} ln ∣ x + 2 ∣

\int \frac{1}{8 ( x - 6 )} d x = \frac{1}{8} ln ∣ x - 6 ∣

= \frac{1}{8} ln ∣ x + 2 ∣ - \frac{1}{8} ln ∣ x - 6 ∣

解答补常数

= \frac{1}{8} ln ∣ x + 2 ∣ - \frac{1}{8} ln ∣ x - 6 ∣ + C

in t e g r a l \frac{1}{( x ^{2} + 4 ) ^{2}}

\int 5 cos^{6} (x) sin^{2} (x) d x

\int 2 x 10 x - x^{2} d x

\int \frac{3 x}{( 3 x ^{2} - 1 ) ^{4}} d x

\int \frac{2 x - 5}{x ( x - 1 ) ^{3}} d x