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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

u置換積分法を適用する

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

部分積分を適用する

= 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) ∣)

代用を戻す

u = arctan (x)

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

簡素化

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}

定数を解答に追加する

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

グラフ

人気の例

integral of sin(x)(2+cos(x))^2

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

integral of (arctan(t))^2

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

integral of cosh(5t-10)

\int cosh (5 t - 10) d t

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

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

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

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