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Beliebt Rechnen >

integral of x(arctan(x))^2

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vereinfache

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (tan(x))/(cos^3(x))

\int \frac{tan ( x )}{cos ^{3} ( x )} d x

(dy)/(y^2-4)=dx

\frac{d y}{y ^{2} - 4} = d x

(\partial)/(\partial x)(-3x)

\frac{\partial}{\partial x} (- 3 x)

integral of 1/(x^2)-1/(xsqrt(x))+5

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

(dy)/(dx)=sqrt(3y)e^{x+7}

\frac{d y}{d x} = 3 y e^{x + 7}