integral of (4x-3)/(x^2+4x+20)

解答

\int \frac{4 x - 3}{x ^{2} + 4 x + 20} d x

2 ln \frac{1}{16} (x + 2)^{2} + 1 - \frac{11}{4} arctan (\frac{1}{4} (x + 2)) + C

求解步骤

\int \frac{4 x - 3}{x ^{2} + 4 x + 20} d x

乘开

\frac{4 x - 3}{x ^{2} + 4 x + 20} : \frac{4 x}{x ^{2} + 4 x + 20} - \frac{3}{x ^{2} + 4 x + 20}

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{4 x}{x ^{2} + 4 x + 20} d x - \int \frac{3}{x ^{2} + 4 x + 20} d x

\int \frac{4 x}{x ^{2} + 4 x + 20} d x = 2 (ln \frac{1}{16} (x + 2)^{2} + 1 - arctan (\frac{1}{4} (x + 2)))

\int \frac{3}{x ^{2} + 4 x + 20} d x = \frac{3}{4} arctan (\frac{1}{4} (x + 2))

= 2 (ln \frac{1}{16} (x + 2)^{2} + 1 - arctan (\frac{1}{4} (x + 2))) - \frac{3}{4} arctan (\frac{1}{4} (x + 2))

化简

2 (ln \frac{1}{16} (x + 2)^{2} + 1 - arctan (\frac{1}{4} (x + 2))) - \frac{3}{4} arctan (\frac{1}{4} (x + 2)) : 2 ln \frac{1}{16} (x + 2)^{2} + 1 - \frac{11}{4} arctan (\frac{1}{4} (x + 2))

= 2 ln \frac{1}{16} (x + 2)^{2} + 1 - \frac{11}{4} arctan (\frac{1}{4} (x + 2))

解答补常数

= 2 ln \frac{1}{16} (x + 2)^{2} + 1 - \frac{11}{4} arctan (\frac{1}{4} (x + 2)) + C

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

\int \frac{4}{9 x ^{2} - 6 x} d x

\int \frac{- cos ( x )}{( 1 + sin ( x ) ) ^{2}} d x

\int \frac{x}{x - 100} d x

\int (y^{3} - 1) e^{x} d x