integral of 1/(x^2)arctan(x)

解答

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

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

求解步骤

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

化简

\frac{1}{x ^{2}} arctan (x) : \frac{arctan ( x )}{x ^{2}}

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

使用分布积分法

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

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

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

化简

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

解答补常数

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

x \to 4 + lim (x - 4)

d er i v a t i v e f (x) = 0.6 x^{1.5}

\frac{d y}{d x} = 9.8 - 0.196 y

y^{^{''}} - 4 y^{^{'}} - 21 y = 0

\int sec^{4} (u) tan (u) d u