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

解答

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

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

求解步骤

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

使用换元积分法

= \int tan^{2} (u) u d u

使用分布积分法

= u (- u + tan (u)) - \int - u + tan (u) d u

\int - u + tan (u) d u = - \frac{u ^{2}}{2} - ln ∣ cos (u) ∣

= u (- u + tan (u)) - (- \frac{u ^{2}}{2} - ln ∣ cos (u) ∣)

u = arctan (x)

代回

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

化简

arctan (x) (- arctan (x) + tan (arctan (x))) - (- \frac{arctan ^{2} ( x )}{2} - ln ∣ cos (arctan (x)) ∣) : - arctan^{2} (x) + x arctan (x) + \frac{1}{2} arctan^{2} (x) - ln 1 + x^{2}

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

解答补常数

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

\int sin (x) (2 + cos (x))^{2} d x

\int (arctan (t))^{2} d t

\int cosh (5 t - 10) d t

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

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