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