integral de θarctan^2(θ)

Solução

\int θ arctan^{2} (θ) d θ

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

Passos da solução

\int θ arctan^{2} (θ) d θ

Aplicar integração por partes

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

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

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

Simplificar

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

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

Adicionar uma constante à solução

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

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