Sr Examen
Integral de sqrt(x)/(2+x^2) dx
Solución
Ha introducido
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___ \/ 2 / | | ___ | \/ x | ------ dx | 2 | 2 + x | / 1
$$\int\limits_{1}^{\sqrt{2}} \frac{\sqrt{x}}{x^{2} + 2}\, dx$$
Integral(sqrt(x)/(2 + x^2), (x, 1, sqrt(2)))
Respuesta (Indefinida)
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/ | | ___ 4 ___ / 4 ___ ___\ 4 ___ / 4 ___ ___\ 4 ___ / ___ 3/4 ___\ 4 ___ / ___ 3/4 ___\ | \/ x \/ 2 *atan\1 + \/ 2 *\/ x / \/ 2 *atan\-1 + \/ 2 *\/ x / \/ 2 *log\x + \/ 2 + 2 *\/ x / \/ 2 *log\x + \/ 2 - 2 *\/ x / | ------ dx = C + --------------------------- + ---------------------------- - --------------------------------- + --------------------------------- | 2 2 2 4 4 | 2 + x | /
$$\int \frac{\sqrt{x}}{x^{2} + 2}\, dx = C + \frac{\sqrt[4]{2} \log{\left(- 2^{\frac{3}{4}} \sqrt{x} + x + \sqrt{2} \right)}}{4} - \frac{\sqrt[4]{2} \log{\left(2^{\frac{3}{4}} \sqrt{x} + x + \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \operatorname{atan}{\left(\sqrt[4]{2} \sqrt{x} - 1 \right)}}{2} + \frac{\sqrt[4]{2} \operatorname{atan}{\left(\sqrt[4]{2} \sqrt{x} + 1 \right)}}{2}$$
Gráfica
Respuesta
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4 ___ / 4 ___\ 4 ___ / 4 ___\ 4 ___ / ___\ 4 ___ / 3/4 ___\ 4 ___ 4 ___ / ___\ 4 ___ / ___ 3/4\
\/ 2 *atan\1 - \/ 2 / \/ 2 *atan\1 + \/ 2 / \/ 2 *log\8 + 8*\/ 2 / \/ 2 *log\4 - 4*2 + 4*\/ 2 / pi*\/ 2 \/ 2 *log\-8 + 8*\/ 2 / \/ 2 *log\4 + 4*\/ 2 + 4*2 /
--------------------- - --------------------- - ---------------------- - ------------------------------- + -------- + ----------------------- + -------------------------------
2 2 4 4 4 4 4
$$- \frac{\sqrt[4]{2} \log{\left(8 + 8 \sqrt{2} \right)}}{4} - \frac{\sqrt[4]{2} \operatorname{atan}{\left(1 + \sqrt[4]{2} \right)}}{2} - \frac{\sqrt[4]{2} \log{\left(- 4 \cdot 2^{\frac{3}{4}} + 4 + 4 \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \operatorname{atan}{\left(1 - \sqrt[4]{2} \right)}}{2} + \frac{\sqrt[4]{2} \log{\left(-8 + 8 \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \log{\left(4 + 4 \sqrt{2} + 4 \cdot 2^{\frac{3}{4}} \right)}}{4} + \frac{\sqrt[4]{2} \pi}{4}$$
=
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4 ___ / 4 ___\ 4 ___ / 4 ___\ 4 ___ / ___\ 4 ___ / 3/4 ___\ 4 ___ 4 ___ / ___\ 4 ___ / ___ 3/4\
\/ 2 *atan\1 - \/ 2 / \/ 2 *atan\1 + \/ 2 / \/ 2 *log\8 + 8*\/ 2 / \/ 2 *log\4 - 4*2 + 4*\/ 2 / pi*\/ 2 \/ 2 *log\-8 + 8*\/ 2 / \/ 2 *log\4 + 4*\/ 2 + 4*2 /
--------------------- - --------------------- - ---------------------- - ------------------------------- + -------- + ----------------------- + -------------------------------
2 2 4 4 4 4 4
$$- \frac{\sqrt[4]{2} \log{\left(8 + 8 \sqrt{2} \right)}}{4} - \frac{\sqrt[4]{2} \operatorname{atan}{\left(1 + \sqrt[4]{2} \right)}}{2} - \frac{\sqrt[4]{2} \log{\left(- 4 \cdot 2^{\frac{3}{4}} + 4 + 4 \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \operatorname{atan}{\left(1 - \sqrt[4]{2} \right)}}{2} + \frac{\sqrt[4]{2} \log{\left(-8 + 8 \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \log{\left(4 + 4 \sqrt{2} + 4 \cdot 2^{\frac{3}{4}} \right)}}{4} + \frac{\sqrt[4]{2} \pi}{4}$$
2^(1/4)*atan(1 - 2^(1/4))/2 - 2^(1/4)*atan(1 + 2^(1/4))/2 - 2^(1/4)*log(8 + 8*sqrt(2))/4 - 2^(1/4)*log(4 - 4*2^(3/4) + 4*sqrt(2))/4 + pi*2^(1/4)/4 + 2^(1/4)*log(-8 + 8*sqrt(2))/4 + 2^(1/4)*log(4 + 4*sqrt(2) + 4*2^(3/4))/4
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.