Sr Examen
Integral de 2x/x*sqrt(2x+1)/sqrt(2x) dx
Solución
Ha introducido
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2 / | | 2*x _________ | ---*\/ 2*x + 1 | x | --------------- dx | _____ | \/ 2*x | / 1
$$\int\limits_{1}^{2} \frac{\frac{2 x}{x} \sqrt{2 x + 1}}{\sqrt{2 x}}\, dx$$
Integral((((2*x)/x)*sqrt(2*x + 1))/sqrt(2*x), (x, 1, 2))
Respuesta (Indefinida)
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/ | // ___ _________ / ___ _________\ \ | 2*x _________ || 2*\/ x *\/ 1/2 + x + acosh\\/ 2 *\/ 1/2 + x / for 2*|1/2 + x| > 1| | ---*\/ 2*x + 1 || | | x || _________ 3/2 | | --------------- dx = C + |< / ___ _________\ I*\/ 1/2 + x 2*I*(1/2 + x) | | _____ ||- I*asin\\/ 2 *\/ 1/2 + x / + ------------- - ---------------- otherwise | | \/ 2*x || ____ ____ | | || \/ -x \/ -x | / \\ /
$$\int \frac{\frac{2 x}{x} \sqrt{2 x + 1}}{\sqrt{2 x}}\, dx = C + \begin{cases} 2 \sqrt{x} \sqrt{x + \frac{1}{2}} + \operatorname{acosh}{\left(\sqrt{2} \sqrt{x + \frac{1}{2}} \right)} & \text{for}\: 2 \left|{x + \frac{1}{2}}\right| > 1 \\- i \operatorname{asin}{\left(\sqrt{2} \sqrt{x + \frac{1}{2}} \right)} - \frac{2 i \left(x + \frac{1}{2}\right)^{\frac{3}{2}}}{\sqrt{- x}} + \frac{i \sqrt{x + \frac{1}{2}}}{\sqrt{- x}} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta
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/ ___ / ___\\ / ___ / ___\\
___ | ____ \/ 2 *acosh\\/ 5 /| ___ | ___ \/ 2 *acosh\\/ 3 /|
\/ 2 *|\/ 10 + ------------------| - \/ 2 *|\/ 3 + ------------------|
\ 2 / \ 2 /
$$- \sqrt{2} \left(\frac{\sqrt{2} \operatorname{acosh}{\left(\sqrt{3} \right)}}{2} + \sqrt{3}\right) + \sqrt{2} \left(\frac{\sqrt{2} \operatorname{acosh}{\left(\sqrt{5} \right)}}{2} + \sqrt{10}\right)$$
=
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/ ___ / ___\\ / ___ / ___\\
___ | ____ \/ 2 *acosh\\/ 5 /| ___ | ___ \/ 2 *acosh\\/ 3 /|
\/ 2 *|\/ 10 + ------------------| - \/ 2 *|\/ 3 + ------------------|
\ 2 / \ 2 /
$$- \sqrt{2} \left(\frac{\sqrt{2} \operatorname{acosh}{\left(\sqrt{3} \right)}}{2} + \sqrt{3}\right) + \sqrt{2} \left(\frac{\sqrt{2} \operatorname{acosh}{\left(\sqrt{5} \right)}}{2} + \sqrt{10}\right)$$
sqrt(2)*(sqrt(10) + sqrt(2)*acosh(sqrt(5))/2) - sqrt(2)*(sqrt(3) + sqrt(2)*acosh(sqrt(3))/2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.