Sr Examen
Integral de 1/(sqrt((y+1)^2-4)) dy
Solución
Ha introducido
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1 / | | 1 | ----------------- dy | ______________ | / 2 | \/ (y + 1) - 4 | / 0
$$\int\limits_{0}^{1} \frac{1}{\sqrt{\left(y + 1\right)^{2} - 4}}\, dy$$
Integral(1/(sqrt((y + 1)^2 - 4)), (y, 0, 1))
Respuesta (Indefinida)
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// | 2| \ / || /1 y\ |(1 + y) | | | || acosh|- + -| for ---------- > 1| | 1 || \2 2/ 4 | | ----------------- dy = C + |< | | ______________ || /1 y\ | | / 2 ||-I*asin|- + -| otherwise | | \/ (y + 1) - 4 || \2 2/ | | \\ / /
$$\int \frac{1}{\sqrt{\left(y + 1\right)^{2} - 4}}\, dy = C + \begin{cases} \operatorname{acosh}{\left(\frac{y}{2} + \frac{1}{2} \right)} & \text{for}\: \frac{\left|{\left(y + 1\right)^{2}}\right|}{4} > 1 \\- i \operatorname{asin}{\left(\frac{y}{2} + \frac{1}{2} \right)} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta
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1 / | | / 2 | | 1 (1 + y) | |---------------------- for -------- > 1 | | _______________ 4 | | / 2 | | / /1 y\ | |2* / -1 + |- + -| | | \/ \2 2/ | < dy | | -I | |--------------------- otherwise | | ______________ | | / 2 | | / /1 y\ | |2* / 1 - |- + -| | | \/ \2 2/ | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{1}{2 \sqrt{\left(\frac{y}{2} + \frac{1}{2}\right)^{2} - 1}} & \text{for}\: \frac{\left(y + 1\right)^{2}}{4} > 1 \\- \frac{i}{2 \sqrt{1 - \left(\frac{y}{2} + \frac{1}{2}\right)^{2}}} & \text{otherwise} \end{cases}\, dy$$
=
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1 / | | / 2 | | 1 (1 + y) | |---------------------- for -------- > 1 | | _______________ 4 | | / 2 | | / /1 y\ | |2* / -1 + |- + -| | | \/ \2 2/ | < dy | | -I | |--------------------- otherwise | | ______________ | | / 2 | | / /1 y\ | |2* / 1 - |- + -| | | \/ \2 2/ | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{1}{2 \sqrt{\left(\frac{y}{2} + \frac{1}{2}\right)^{2} - 1}} & \text{for}\: \frac{\left(y + 1\right)^{2}}{4} > 1 \\- \frac{i}{2 \sqrt{1 - \left(\frac{y}{2} + \frac{1}{2}\right)^{2}}} & \text{otherwise} \end{cases}\, dy$$
Integral(Piecewise((1/(2*sqrt(-1 + (1/2 + y/2)^2)), (1 + y)^2/4 > 1), (-i/(2*sqrt(1 - (1/2 + y/2)^2)), True)), (y, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.