integral of (3x^2+5x+3)/((x^2+1)^2)

解答

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

3 arctan (x) - \frac{5}{2 ( x ^{2} + 1 )} + C

求解步骤

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

将

\frac{3 x ^{2} + 5 x + 3}{( x ^{2} + 1 ) ^{2}}

用部份分式展开:

\frac{3}{x ^{2} + 1} + \frac{5 x}{( x ^{2} + 1 ) ^{2}}

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

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

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

= 3 arctan (x) - \frac{5}{2 ( x ^{2} + 1 )}

解答补常数

= 3 arctan (x) - \frac{5}{2 ( x ^{2} + 1 )} + C

d er i v a t i v e f (x) = 5 x^{2} tan (1 - x)

d er i v a t i v e x (x^{2} + 7)^{4}

t an g e n t 2 x^{3} - 2 x^{2} + 1

x \to 7 + lim (\frac{3}{x - 7})

\int_{5}^{\infty} \frac{1}{x ^{2} - 4} d x