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

解答

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

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

求解步骤

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

将

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

用部份分式展开:

\frac{x - 1}{2 ( x ^{2} + 1 )} + \frac{1}{2 ( x + 1 )}

= 3 \cdot \int \frac{x - 1}{2 ( x ^{2} + 1 )} + \frac{1}{2 ( x + 1 )} 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

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

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

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

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

解答补常数

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

d er i v a t i v e sin (x) + \frac{4}{3} cot (x)

\int \frac{1}{x ^{2} - 7 x + 12} d x

x y^{2} \frac{d y}{d x} = y^{3} - 3 x^{3}, y (1) = 5

y^{^{'}} = 0.1 \cdot y - 250000

\int 5 cos (π x - 3) d x