integral de (x^2+x)/(sqrt(1-x^2))

Solução

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

\frac{1}{2} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))) - 1 - x^{2} + C

Passos da solução

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

Expandir

\frac{x ^{2} + x}{1 - x ^{2}} : \frac{x ^{2}}{1 - x ^{2}} + \frac{x}{1 - x ^{2}}

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

Aplicar a regra da soma:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{x ^{2}}{1 - x ^{2}} d x = \frac{1}{2} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x)))

\int \frac{x}{1 - x ^{2}} d x = - 1 - x^{2}

= \frac{1}{2} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))) - 1 - x^{2}

Adicionar uma constante à solução

= \frac{1}{2} (arcsin (x) - \frac{1}{2} sin (2 arcsin (x))) - 1 - x^{2} + C

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