Sr Examen
Integral de 1/(x(1+x^1/3)^3) dx
Solución
Ha introducido
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1 / | | 1 | -------------- dx | 3 | / 3 ___\ | x*\1 + \/ x / | / 0
$$\int\limits_{0}^{1} \frac{1}{x \left(\sqrt[3]{x} + 1\right)^{3}}\, dx$$
Integral(1/(x*(1 + x^(1/3))^3), (x, 0, 1))
Respuesta (Indefinida)
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/ | 2/3 / 3 ___\ 2/3 / 3 ___\ 4/3 / 3 ___\ 2/3 4/3 | 1 6*x 9*x 12*x*log\1 + \/ x / 6*x *log\1 + \/ x / 6*x *log\1 + \/ x / 2*x *log(x) 2*x *log(x) 4*x*log(x) | -------------- dx = C + --------------------- + --------------------- - --------------------- - --------------------- - --------------------- + --------------------- + --------------------- + --------------------- | 3 2/3 4/3 2/3 4/3 2/3 4/3 2/3 4/3 2/3 4/3 2/3 4/3 2/3 4/3 2/3 4/3 | / 3 ___\ 2*x + 2*x + 4*x 2*x + 2*x + 4*x 2*x + 2*x + 4*x 2*x + 2*x + 4*x 2*x + 2*x + 4*x 2*x + 2*x + 4*x 2*x + 2*x + 4*x 2*x + 2*x + 4*x | x*\1 + \/ x / | /
$$\int \frac{1}{x \left(\sqrt[3]{x} + 1\right)^{3}}\, dx = C + \frac{2 x^{\frac{4}{3}} \log{\left(x \right)}}{2 x^{\frac{4}{3}} + 2 x^{\frac{2}{3}} + 4 x} - \frac{6 x^{\frac{4}{3}} \log{\left(\sqrt[3]{x} + 1 \right)}}{2 x^{\frac{4}{3}} + 2 x^{\frac{2}{3}} + 4 x} + \frac{2 x^{\frac{2}{3}} \log{\left(x \right)}}{2 x^{\frac{4}{3}} + 2 x^{\frac{2}{3}} + 4 x} - \frac{6 x^{\frac{2}{3}} \log{\left(\sqrt[3]{x} + 1 \right)}}{2 x^{\frac{4}{3}} + 2 x^{\frac{2}{3}} + 4 x} + \frac{9 x^{\frac{2}{3}}}{2 x^{\frac{4}{3}} + 2 x^{\frac{2}{3}} + 4 x} + \frac{4 x \log{\left(x \right)}}{2 x^{\frac{4}{3}} + 2 x^{\frac{2}{3}} + 4 x} - \frac{12 x \log{\left(\sqrt[3]{x} + 1 \right)}}{2 x^{\frac{4}{3}} + 2 x^{\frac{2}{3}} + 4 x} + \frac{6 x}{2 x^{\frac{4}{3}} + 2 x^{\frac{2}{3}} + 4 x}$$
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.