Sr Examen
Integral de x/(x^3+16) dx
Solución
Ha introducido
[src]
10 / | | x | ------- dx | 3 | x + 16 | / 0
$$\int\limits_{0}^{10} \frac{x}{x^{3} + 16}\, dx$$
Integral(x/(x^3 + 16), (x, 0, 10))
Respuesta (Indefinida)
[src]
/ ___ 2/3 ___\ / 2/3 ___ | \/ 3 x*2 *\/ 3 | | 2/3 / 3 ___\ 2/3 / 2 2/3 3 ___\ 2 *\/ 3 *atan|- ----- + ------------| | x 2 *log\x + 2*\/ 2 / 2 *log\x + 4*2 - 2*x*\/ 2 / \ 3 6 / | ------- dx = C - --------------------- + --------------------------------- + --------------------------------------- | 3 12 24 12 | x + 16 | /
$$\int \frac{x}{x^{3} + 16}\, dx = C - \frac{2^{\frac{2}{3}} \log{\left(x + 2 \sqrt[3]{2} \right)}}{12} + \frac{2^{\frac{2}{3}} \log{\left(x^{2} - 2 \sqrt[3]{2} x + 4 \cdot 2^{\frac{2}{3}} \right)}}{24} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{6} - \frac{\sqrt{3}}{3} \right)}}{12}$$
Gráfica
Respuesta
[src]
/ ___ 2/3 ___\
2/3 ___ |\/ 3 5*2 *\/ 3 |
2/3 / 3 ___\ 2/3 / 2/3\ 2/3 / 3 ___\ 2/3 / 3 ___ 2/3\ 2 *\/ 3 *atan|----- - ------------| 2/3 ___
2 *log\10 + 2*\/ 2 / 2 *log\4*2 / 2 *log\2*\/ 2 / 2 *log\100 - 20*\/ 2 + 4*2 / \ 3 3 / pi*2 *\/ 3
- ---------------------- - ---------------- + ----------------- + --------------------------------- - ------------------------------------- + -------------
12 24 12 24 12 72
$$- \frac{2^{\frac{2}{3}} \log{\left(2 \sqrt[3]{2} + 10 \right)}}{12} - \frac{2^{\frac{2}{3}} \log{\left(4 \cdot 2^{\frac{2}{3}} \right)}}{24} + \frac{2^{\frac{2}{3}} \sqrt{3} \pi}{72} + \frac{2^{\frac{2}{3}} \log{\left(2 \sqrt[3]{2} \right)}}{12} + \frac{2^{\frac{2}{3}} \log{\left(- 20 \sqrt[3]{2} + 4 \cdot 2^{\frac{2}{3}} + 100 \right)}}{24} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(- \frac{5 \cdot 2^{\frac{2}{3}} \sqrt{3}}{3} + \frac{\sqrt{3}}{3} \right)}}{12}$$
=
=
/ ___ 2/3 ___\
2/3 ___ |\/ 3 5*2 *\/ 3 |
2/3 / 3 ___\ 2/3 / 2/3\ 2/3 / 3 ___\ 2/3 / 3 ___ 2/3\ 2 *\/ 3 *atan|----- - ------------| 2/3 ___
2 *log\10 + 2*\/ 2 / 2 *log\4*2 / 2 *log\2*\/ 2 / 2 *log\100 - 20*\/ 2 + 4*2 / \ 3 3 / pi*2 *\/ 3
- ---------------------- - ---------------- + ----------------- + --------------------------------- - ------------------------------------- + -------------
12 24 12 24 12 72
$$- \frac{2^{\frac{2}{3}} \log{\left(2 \sqrt[3]{2} + 10 \right)}}{12} - \frac{2^{\frac{2}{3}} \log{\left(4 \cdot 2^{\frac{2}{3}} \right)}}{24} + \frac{2^{\frac{2}{3}} \sqrt{3} \pi}{72} + \frac{2^{\frac{2}{3}} \log{\left(2 \sqrt[3]{2} \right)}}{12} + \frac{2^{\frac{2}{3}} \log{\left(- 20 \sqrt[3]{2} + 4 \cdot 2^{\frac{2}{3}} + 100 \right)}}{24} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(- \frac{5 \cdot 2^{\frac{2}{3}} \sqrt{3}}{3} + \frac{\sqrt{3}}{3} \right)}}{12}$$
-2^(2/3)*log(10 + 2*2^(1/3))/12 - 2^(2/3)*log(4*2^(2/3))/24 + 2^(2/3)*log(2*2^(1/3))/12 + 2^(2/3)*log(100 - 20*2^(1/3) + 4*2^(2/3))/24 - 2^(2/3)*sqrt(3)*atan(sqrt(3)/3 - 5*2^(2/3)*sqrt(3)/3)/12 + pi*2^(2/3)*sqrt(3)/72
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.