integral of (29x^4+29x^3+2x+1)/(x^2+x)

解答

\int \frac{29 x ^{4} + 29 x ^{3} + 2 x + 1}{x ^{2} + x} d x

29 x + ln ∣ x + 1 ∣ + ln ∣ x ∣ + \frac{29}{3} (x + 1)^{3} + \frac{29}{2} (x + 1)^{2} - \frac{87}{2} (x + 1)^{2} + 29 + C

求解步骤

\int \frac{29 x ^{4} + 29 x ^{3} + 2 x + 1}{x ^{2} + x} d x

乘开

\frac{29 x ^{4} + 29 x ^{3} + 2 x + 1}{x ^{2} + x} : \frac{29 x ^{4}}{x ^{2} + x} + \frac{29 x ^{3}}{x ^{2} + x} + \frac{2 x}{x ^{2} + x} + \frac{1}{x ^{2} + 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{29 x ^{4}}{x ^{2} + x} d x + \int \frac{29 x ^{3}}{x ^{2} + x} d x + \int \frac{2 x}{x ^{2} + x} d x + \int \frac{1}{x ^{2} + x} d x

\int \frac{29 x ^{4}}{x ^{2} + x} d x = 29 (\frac{( x + 1 ) ^{3}}{3} - \frac{3 ( x + 1 ) ^{2}}{2} + 3 (x + 1) - ln ∣ x + 1 ∣)

\int \frac{29 x ^{3}}{x ^{2} + x} d x = 29 (\frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + ln ∣ x + 1 ∣)

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

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

= 29 (\frac{( x + 1 ) ^{3}}{3} - \frac{3 ( x + 1 ) ^{2}}{2} + 3 (x + 1) - ln ∣ x + 1 ∣) + 29 (\frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + ln ∣ x + 1 ∣) + 2 ln ∣ x + 1 ∣ + ln ∣ x ∣ - ln ∣ x + 1 ∣

化简

29 (\frac{( x + 1 ) ^{3}}{3} - \frac{3 ( x + 1 ) ^{2}}{2} + 3 (x + 1) - ln ∣ x + 1 ∣) + 29 (\frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + ln ∣ x + 1 ∣) + 2 ln ∣ x + 1 ∣ + ln ∣ x ∣ - ln ∣ x + 1 ∣ : 29 x + ln ∣ x + 1 ∣ + ln ∣ x ∣ + \frac{29}{3} (x + 1)^{3} + \frac{29}{2} (x + 1)^{2} - \frac{87}{2} (x + 1)^{2} + 29

= 29 x + ln ∣ x + 1 ∣ + ln ∣ x ∣ + \frac{29}{3} (x + 1)^{3} + \frac{29}{2} (x + 1)^{2} - \frac{87}{2} (x + 1)^{2} + 29

解答补常数

= 29 x + ln ∣ x + 1 ∣ + ln ∣ x ∣ + \frac{29}{3} (x + 1)^{3} + \frac{29}{2} (x + 1)^{2} - \frac{87}{2} (x + 1)^{2} + 29 + C

\frac{d}{d x} y = (sin (\frac{x}{2}) - cos (\frac{x}{2}))^{2}

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

\frac{\partial}{\partial x} (a r cot (\frac{y}{x}))

\int_{0}^{1} x (4 3 x + 7 4 x) d x

d er i v a t i v e y = e^{2 e^{x}}