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integral de (4x^2)/(x^4-1)

Solução

\int \frac{4 x ^{2}}{x ^{4} - 1} d x

Solução

2 arctan (x) - ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + C

Passos da solução

\int \frac{4 x ^{2}}{x ^{4} - 1} d x

Remover a constante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int \frac{x ^{2}}{x ^{4} - 1} d x

Tirar a fração parcial de

\frac{x ^{2}}{x ^{4} - 1} : \frac{1}{2 ( x ^{2} + 1 )} - \frac{1}{4 ( x + 1 )} + \frac{1}{4 ( x - 1 )}

= 4 \cdot \int \frac{1}{2 ( x ^{2} + 1 )} - \frac{1}{4 ( x + 1 )} + \frac{1}{4 ( x - 1 )} 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

= 4 (\int \frac{1}{2 ( x ^{2} + 1 )} d x - \int \frac{1}{4 ( x + 1 )} d x + \int \frac{1}{4 ( x - 1 )} d x)

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

\int \frac{1}{4 ( x + 1 )} d x = \frac{1}{4} ln ∣ x + 1 ∣

\int \frac{1}{4 ( x - 1 )} d x = \frac{1}{4} ln ∣ x - 1 ∣

= 4 (\frac{1}{2} arctan (x) - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣)

Simplificar

4 (\frac{1}{2} arctan (x) - \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{4} ln ∣ x - 1 ∣) : 2 arctan (x) - ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣

= 2 arctan (x) - ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣

Adicionar uma constante à solução

= 2 arctan (x) - ln ∣ x + 1 ∣ + ln ∣ x - 1 ∣ + C

Exemplos populares

integral de (x^3)/(sqrt(x^4+12))

\int \frac{x ^{3}}{x ^{4} + 12} d x

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

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

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

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

integral de 2/(x^2+25)

\int \frac{2}{x ^{2} + 25} d x

integral de 4x^3sin(x^4)

\int 4 x^{3} sin (x^{4}) d x