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

解答

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

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

求解步骤

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

乘开

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

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

\int \frac{x}{1 + x ^{2}} d x = \frac{1}{2} ln 1 + x^{2}

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

解答补常数

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

a re a x^{2} + 10, 9 x, 0

\int \frac{( 9 - y ^{2} ) ^{2}}{2} - y (9 - y^{2}) d y

\int y + 4 d y

\frac{d}{d x} (\frac{2 x}{x ^{4} + 1})

\int_{1}^{4} - 10 t ln (t) d t