integral of (3x+1)/(x^2+2x+2)

解答

\int \frac{3 x + 1}{x ^{2} + 2 x + 2} d x

\frac{3}{2} ln x^{2} + 2 x + 2 - 2 arctan (x + 1) + C

求解步骤

\int \frac{3 x + 1}{x ^{2} + 2 x + 2} d x

乘开

\frac{3 x + 1}{x ^{2} + 2 x + 2} : \frac{3 x}{x ^{2} + 2 x + 2} + \frac{1}{x ^{2} + 2 x + 2}

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{3 x}{x ^{2} + 2 x + 2} d x + \int \frac{1}{x ^{2} + 2 x + 2} d x

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

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

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

化简

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

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

解答补常数

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

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

\int \frac{x ^{4} - 8 x ^{2} + 5 x}{x ^{2}} d x

\int 9 + e^{2 x} - \frac{5}{x ^{3}} d x

\int (x^{3} + 4 x^{2} - 5) d x

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