integral of (2x^2+x)/(1+sqrt(1-x^2))

解答

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

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

求解步骤

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

使用分布积分法

= - 2 x + 2 x - x^{2} + 1 + 2 x^{2} arcsin (x) - 1 + 1 - x^{2} + x arcsin (x) - \int - 4 + 4 1 - x^{2} + 4 x arcsin (x) - \frac{1}{x} + \frac{1 - x ^{2}}{x} + arcsin (x) d x

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

= - 2 x + 2 x - x^{2} + 1 + 2 x^{2} arcsin (x) - 1 + 1 - x^{2} + x arcsin (x) - (2 x^{2} arcsin (x) - 4 x + x arcsin (x) + \frac{3}{2} sin (2 arcsin (x)) - \frac{1}{2} ln - x^{2} + 1 + 1 + \frac{1}{2} ln - x^{2} + 1 - 1 + arcsin (x) - ln ∣ x ∣ + 2 - x^{2} + 1)

化简

- 2 x + 2 x - x^{2} + 1 + 2 x^{2} arcsin (x) - 1 + 1 - x^{2} + x arcsin (x) - (2 x^{2} arcsin (x) - 4 x + x arcsin (x) + \frac{3}{2} sin (2 arcsin (x)) - \frac{1}{2} ln - x^{2} + 1 + 1 + \frac{1}{2} ln - x^{2} + 1 - 1 + arcsin (x) - ln ∣ x ∣ + 2 - x^{2} + 1) : 2 x + 2 x - x^{2} + 1 - \frac{3}{2} sin (2 arcsin (x)) + \frac{1}{2} ln - x^{2} + 1 + 1 - \frac{1}{2} ln - x^{2} + 1 - 1 - - x^{2} + 1 - arcsin (x) + ln ∣ x ∣ - 1

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

解答补常数

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

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

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