Sr Examen
Integral de 4-(sqrt(16-((x-4)^2))) dx
Solución
Ha introducido
[src]
1 / | | / _______________\ | | / 2 | | \4 - \/ 16 - (x - 4) / dx | / 0
$$\int\limits_{0}^{1} \left(4 - \sqrt{16 - \left(x - 4\right)^{2}}\right)\, dx$$
Integral(4 - sqrt(16 - (x - 4)^2), (x, 0, 1))
Solución detallada
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Integramos término a término:
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La integral de las constantes tienen esta constante multiplicada por la variable de integración:
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La integral del producto de una función por una constante es la constante por la integral de esta función:
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No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
Por lo tanto, el resultado es:
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El resultado es:
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Ahora simplificar:
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Añadimos la constante de integración:
Respuesta:
Respuesta (Indefinida)
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// 3 | 2| \
|| / x\ I*(-4 + x) 8*I*(-4 + x) |(-4 + x) | |
/ ||- 8*I*acosh|-1 + -| + ---------------------- - -------------------- for ----------- > 1|
| || \ 4/ _________________ _________________ 16 |
| / _______________\ || / 2 / 2 |
| | / 2 | || 2*\/ -16 + (-4 + x) \/ -16 + (-4 + x) |
| \4 - \/ 16 - (x - 4) / dx = C - |< | + 4*x
| || 3 |
/ || / x\ 8*(-4 + x) (-4 + x) |
|| 8*asin|-1 + -| + ------------------- - --------------------- otherwise |
|| \ 4/ ________________ ________________ |
|| / 2 / 2 |
\\ \/ 16 - (-4 + x) 2*\/ 16 - (-4 + x) /
$$\int \left(4 - \sqrt{16 - \left(x - 4\right)^{2}}\right)\, dx = C + 4 x - \begin{cases} \frac{i \left(x - 4\right)^{3}}{2 \sqrt{\left(x - 4\right)^{2} - 16}} - \frac{8 i \left(x - 4\right)}{\sqrt{\left(x - 4\right)^{2} - 16}} - 8 i \operatorname{acosh}{\left(\frac{x}{4} - 1 \right)} & \text{for}\: \frac{\left|{\left(x - 4\right)^{2}}\right|}{16} > 1 \\8 \operatorname{asin}{\left(\frac{x}{4} - 1 \right)} - \frac{\left(x - 4\right)^{3}}{2 \sqrt{16 - \left(x - 4\right)^{2}}} + \frac{8 \left(x - 4\right)}{\sqrt{16 - \left(x - 4\right)^{2}}} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta
[src]
1
/
|
| / 2 3 2
| | 2*I 8*I 3*I*(-4 + x) 8*I*(-4 + x)*(4 - x) I*(-4 + x) *(4 - x) (-4 + x)
| |--------------------- + -------------------- - ---------------------- + -------------------- - ---------------------- for --------- > 1
| | ________________ _________________ _________________ 3/2 3/2 16
| | / 2 / 2 / 2 / 2\ / 2\
| | / / x\ \/ -16 + (-4 + x) 2*\/ -16 + (-4 + x) \-16 + (-4 + x) / 2*\-16 + (-4 + x) /
| | / -1 + |-1 + -|
| |\/ \ 4/
4 + | < dx
| | 4 2 2
| | 8 2 (-4 + x) 8*(-4 + x) 3*(-4 + x)
| | - ------------------- - -------------------- + --------------------- - ------------------- + --------------------- otherwise
| | ________________ _______________ 3/2 3/2 ________________
| | / 2 / 2 / 2\ / 2\ / 2
| | \/ 16 - (-4 + x) / / x\ 2*\16 - (-4 + x) / \16 - (-4 + x) / 2*\/ 16 - (-4 + x)
| | / 1 - |-1 + -|
| \ \/ \ 4/
|
/
0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i \left(4 - x\right) \left(x - 4\right)^{3}}{2 \left(\left(x - 4\right)^{2} - 16\right)^{\frac{3}{2}}} + \frac{8 i \left(4 - x\right) \left(x - 4\right)}{\left(\left(x - 4\right)^{2} - 16\right)^{\frac{3}{2}}} - \frac{3 i \left(x - 4\right)^{2}}{2 \sqrt{\left(x - 4\right)^{2} - 16}} + \frac{8 i}{\sqrt{\left(x - 4\right)^{2} - 16}} + \frac{2 i}{\sqrt{\left(\frac{x}{4} - 1\right)^{2} - 1}} & \text{for}\: \frac{\left(x - 4\right)^{2}}{16} > 1 \\\frac{3 \left(x - 4\right)^{2}}{2 \sqrt{16 - \left(x - 4\right)^{2}}} - \frac{8}{\sqrt{16 - \left(x - 4\right)^{2}}} + \frac{\left(x - 4\right)^{4}}{2 \left(16 - \left(x - 4\right)^{2}\right)^{\frac{3}{2}}} - \frac{8 \left(x - 4\right)^{2}}{\left(16 - \left(x - 4\right)^{2}\right)^{\frac{3}{2}}} - \frac{2}{\sqrt{1 - \left(\frac{x}{4} - 1\right)^{2}}} & \text{otherwise} \end{cases}\, dx + 4$$
=
=
1
/
|
| / 2 3 2
| | 2*I 8*I 3*I*(-4 + x) 8*I*(-4 + x)*(4 - x) I*(-4 + x) *(4 - x) (-4 + x)
| |--------------------- + -------------------- - ---------------------- + -------------------- - ---------------------- for --------- > 1
| | ________________ _________________ _________________ 3/2 3/2 16
| | / 2 / 2 / 2 / 2\ / 2\
| | / / x\ \/ -16 + (-4 + x) 2*\/ -16 + (-4 + x) \-16 + (-4 + x) / 2*\-16 + (-4 + x) /
| | / -1 + |-1 + -|
| |\/ \ 4/
4 + | < dx
| | 4 2 2
| | 8 2 (-4 + x) 8*(-4 + x) 3*(-4 + x)
| | - ------------------- - -------------------- + --------------------- - ------------------- + --------------------- otherwise
| | ________________ _______________ 3/2 3/2 ________________
| | / 2 / 2 / 2\ / 2\ / 2
| | \/ 16 - (-4 + x) / / x\ 2*\16 - (-4 + x) / \16 - (-4 + x) / 2*\/ 16 - (-4 + x)
| | / 1 - |-1 + -|
| \ \/ \ 4/
|
/
0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i \left(4 - x\right) \left(x - 4\right)^{3}}{2 \left(\left(x - 4\right)^{2} - 16\right)^{\frac{3}{2}}} + \frac{8 i \left(4 - x\right) \left(x - 4\right)}{\left(\left(x - 4\right)^{2} - 16\right)^{\frac{3}{2}}} - \frac{3 i \left(x - 4\right)^{2}}{2 \sqrt{\left(x - 4\right)^{2} - 16}} + \frac{8 i}{\sqrt{\left(x - 4\right)^{2} - 16}} + \frac{2 i}{\sqrt{\left(\frac{x}{4} - 1\right)^{2} - 1}} & \text{for}\: \frac{\left(x - 4\right)^{2}}{16} > 1 \\\frac{3 \left(x - 4\right)^{2}}{2 \sqrt{16 - \left(x - 4\right)^{2}}} - \frac{8}{\sqrt{16 - \left(x - 4\right)^{2}}} + \frac{\left(x - 4\right)^{4}}{2 \left(16 - \left(x - 4\right)^{2}\right)^{\frac{3}{2}}} - \frac{8 \left(x - 4\right)^{2}}{\left(16 - \left(x - 4\right)^{2}\right)^{\frac{3}{2}}} - \frac{2}{\sqrt{1 - \left(\frac{x}{4} - 1\right)^{2}}} & \text{otherwise} \end{cases}\, dx + 4$$
4 + Integral(Piecewise((2*i/sqrt(-1 + (-1 + x/4)^2) + 8*i/sqrt(-16 + (-4 + x)^2) - 3*i*(-4 + x)^2/(2*sqrt(-16 + (-4 + x)^2)) + 8*i*(-4 + x)*(4 - x)/(-16 + (-4 + x)^2)^(3/2) - i*(-4 + x)^3*(4 - x)/(2*(-16 + (-4 + x)^2)^(3/2)), (-4 + x)^2/16 > 1), (-8/sqrt(16 - (-4 + x)^2) - 2/sqrt(1 - (-1 + x/4)^2) + (-4 + x)^4/(2*(16 - (-4 + x)^2)^(3/2)) - 8*(-4 + x)^2/(16 - (-4 + x)^2)^(3/2) + 3*(-4 + x)^2/(2*sqrt(16 - (-4 + x)^2)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.