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Popular Cálculo >

integral de 0 a 7pi de sqrt(x^4+4x^2)

Solução

\int_{0}^{7 π} x^{4} + 4 x^{2} d x

Solução

\frac{( 49 π ^{2} + 4 ) ^{\frac{3}{2}} - 8}{3}

+1

Decimal

3586.45741 \dots

Passos da solução

\int_{0}^{7 π} x^{4} + 4 x^{2} d x

Multiplicar pelo conjugado de

x^{4} + 4 x^{2} : \frac{x ^{8} - 16 x ^{4}}{x ^{4} - 4 x ^{2}}

= \int_{0}^{7 π} \frac{x ^{8} - 16 x ^{4}}{x ^{4} - 4 x ^{2}} d x

Encontrar os pontos não definidos (singularidades):

x = 2

Se existe

b, a < b < c, f (b) =

indefinida

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{0}^{2} \frac{x ^{8} - 16 x ^{4}}{x ^{4} - 4 x ^{2}} d x + \int_{2}^{7 π} \frac{x ^{8} - 16 x ^{4}}{x ^{4} - 4 x ^{2}} d x

\int_{0}^{2} \frac{x ^{8} - 16 x ^{4}}{x ^{4} - 4 x ^{2}} d x = \frac{16 2 - 8}{3}

\int_{2}^{7 π} \frac{x ^{8} - 16 x ^{4}}{x ^{4} - 4 x ^{2}} d x = \frac{( 49 π ^{2} + 4 ) ^{\frac{3}{2}} - 16 2}{3}

= \frac{16 2 - 8}{3} + \frac{( 49 π ^{2} + 4 ) ^{\frac{3}{2}} - 16 2}{3}

Simplificar

\frac{16 2 - 8}{3} + \frac{( 49 π ^{2} + 4 ) ^{\frac{3}{2}} - 16 2}{3} : \frac{( 49 π ^{2} + 4 ) ^{\frac{3}{2}} - 8}{3}

= \frac{( 49 π ^{2} + 4 ) ^{\frac{3}{2}} - 8}{3}

Gráfico

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