Integral de 1/sqrt(c-1/2*x^2+1/6*a*x3) dx
Solución
Solución detallada
Vuelva a escribir el integrando:
1 a 6 x 3 + ( c − x 2 2 ) = 6 a x 3 + 6 c − 3 x 2 \frac{1}{\sqrt{\frac{a}{6} x_{3} + \left(c - \frac{x^{2}}{2}\right)}} = \frac{\sqrt{6}}{\sqrt{a x_{3} + 6 c - 3 x^{2}}} 6 a x 3 + ( c − 2 x 2 ) 1 = a x 3 + 6 c − 3 x 2 6
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 6 a x 3 + 6 c − 3 x 2 d x = 6 ∫ 1 a x 3 + 6 c − 3 x 2 d x \int \frac{\sqrt{6}}{\sqrt{a x_{3} + 6 c - 3 x^{2}}}\, dx = \sqrt{6} \int \frac{1}{\sqrt{a x_{3} + 6 c - 3 x^{2}}}\, dx ∫ a x 3 + 6 c − 3 x 2 6 d x = 6 ∫ a x 3 + 6 c − 3 x 2 1 d x
PieceweseRule(subfunctions=[(ConstantTimesRule(constant=1/sqrt(a*x3 + 6*c), other=1/sqrt(-3*x**2/(a*x3 + 6*c) + 1), substep=URule(u_var=_u, u_func=sqrt(3)*x*sqrt(1/(a*x3 + 6*c)), constant=sqrt(a*x3/3 + 2*c), substep=ConstantTimesRule(constant=sqrt(a*x3/3 + 2*c), other=1/sqrt(1 - _u**2), substep=ArcsinRule(context=1/sqrt(1 - _u**2), symbol=_u), context=sqrt(a*x3/3 + 2*c)/sqrt(1 - _u**2), symbol=x), context=1/sqrt(-3*x**2/(a*x3 + 6*c) + 1), symbol=x), context=1/sqrt(a*x3 + 6*c - 3*x**2), symbol=x), a*x3 + 6*c > 0)], context=1/sqrt(a*x3 + 6*c - 3*x**2), symbol=x)
Por lo tanto, el resultado es: 6 ( { a x 3 3 + 2 c asin ( 3 x 1 a x 3 + 6 c ) a x 3 + 6 c for a x 3 + 6 c > 0 ) \sqrt{6} \left(\begin{cases} \frac{\sqrt{\frac{a x_{3}}{3} + 2 c} \operatorname{asin}{\left(\sqrt{3} x \sqrt{\frac{1}{a x_{3} + 6 c}} \right)}}{\sqrt{a x_{3} + 6 c}} & \text{for}\: a x_{3} + 6 c > 0 \end{cases}\right) 6 ( { a x 3 + 6 c 3 a x 3 + 2 c asin ( 3 x a x 3 + 6 c 1 ) for a x 3 + 6 c > 0 )
Ahora simplificar:
{ 2 asin ( 3 x 1 a x 3 + 6 c ) for a x 3 + 6 c > 0 \begin{cases} \sqrt{2} \operatorname{asin}{\left(\sqrt{3} x \sqrt{\frac{1}{a x_{3} + 6 c}} \right)} & \text{for}\: a x_{3} + 6 c > 0 \end{cases} { 2 asin ( 3 x a x 3 + 6 c 1 ) for a x 3 + 6 c > 0
Añadimos la constante de integración:
{ 2 asin ( 3 x 1 a x 3 + 6 c ) for a x 3 + 6 c > 0 + c o n s t a n t \begin{cases} \sqrt{2} \operatorname{asin}{\left(\sqrt{3} x \sqrt{\frac{1}{a x_{3} + 6 c}} \right)} & \text{for}\: a x_{3} + 6 c > 0 \end{cases}+ \mathrm{constant} { 2 asin ( 3 x a x 3 + 6 c 1 ) for a x 3 + 6 c > 0 + constant
Respuesta:
{ 2 asin ( 3 x 1 a x 3 + 6 c ) for a x 3 + 6 c > 0 + c o n s t a n t \begin{cases} \sqrt{2} \operatorname{asin}{\left(\sqrt{3} x \sqrt{\frac{1}{a x_{3} + 6 c}} \right)} & \text{for}\: a x_{3} + 6 c > 0 \end{cases}+ \mathrm{constant} { 2 asin ( 3 x a x 3 + 6 c 1 ) for a x 3 + 6 c > 0 + constant
Respuesta (Indefinida)
[src]
/ // ____________ / ____________\ \
| || / a*x3 | ___ / 1 | |
| 1 ___ || / 2*c + ---- *asin|x*\/ 3 * / ---------- | |
| -------------------- dx = C + \/ 6 *|<\/ 3 \ \/ 6*c + a*x3 / |
| _______________ ||----------------------------------------------- for 6*c + a*x3 > 0|
| / 2 || ____________ |
| / x a \\ \/ 6*c + a*x3 /
| / c - -- + -*x3
| \/ 2 6
|
/
∫ 1 a 6 x 3 + ( c − x 2 2 ) d x = C + 6 ( { a x 3 3 + 2 c asin ( 3 x 1 a x 3 + 6 c ) a x 3 + 6 c for a x 3 + 6 c > 0 ) \int \frac{1}{\sqrt{\frac{a}{6} x_{3} + \left(c - \frac{x^{2}}{2}\right)}}\, dx = C + \sqrt{6} \left(\begin{cases} \frac{\sqrt{\frac{a x_{3}}{3} + 2 c} \operatorname{asin}{\left(\sqrt{3} x \sqrt{\frac{1}{a x_{3} + 6 c}} \right)}}{\sqrt{a x_{3} + 6 c}} & \text{for}\: a x_{3} + 6 c > 0 \end{cases}\right) ∫ 6 a x 3 + ( c − 2 x 2 ) 1 d x = C + 6 ( { a x 3 + 6 c 3 a x 3 + 2 c asin ( 3 x a x 3 + 6 c 1 ) for a x 3 + 6 c > 0 )
1
/
|
| / 2
| | -I x
| |--------------------------------------------------------------- for ------------ > 1
| | _____________________________ | a*x3|
| | / 2 ______________________ 2*|c + ----|
| | / x / / a*x3\ | 6 |
| | / -1 + ---------------------- * / polar_lift|c + ----|
| | / / a*x3\ \/ \ 6 /
| | / 2*polar_lift|c + ----|
| |\/ \ 6 /
| < dx
| | 1
| |-------------------------------------------------------------- otherwise
| | ____________________________
| | / 2 ______________________
| | / x / / a*x3\
| | / 1 - ---------------------- * / polar_lift|c + ----|
| | / / a*x3\ \/ \ 6 /
| | / 2*polar_lift|c + ----|
| |\/ \ 6 /
| \
|
/
0
∫ 0 1 { − i x 2 2 polar_lift ( a x 3 6 + c ) − 1 polar_lift ( a x 3 6 + c ) for x 2 2 ∣ a x 3 6 + c ∣ > 1 1 − x 2 2 polar_lift ( a x 3 6 + c ) + 1 polar_lift ( a x 3 6 + c ) otherwise d x \int\limits_{0}^{1} \begin{cases} - \frac{i}{\sqrt{\frac{x^{2}}{2 \operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}}} & \text{for}\: \frac{x^{2}}{2 \left|{\frac{a x_{3}}{6} + c}\right|} > 1 \\\frac{1}{\sqrt{- \frac{x^{2}}{2 \operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}}} & \text{otherwise} \end{cases}\, dx 0 ∫ 1 ⎩ ⎨ ⎧ − 2 polar_lift ( 6 a x 3 + c ) x 2 − 1 polar_lift ( 6 a x 3 + c ) i − 2 polar_lift ( 6 a x 3 + c ) x 2 + 1 polar_lift ( 6 a x 3 + c ) 1 for 2 ∣ 6 a x 3 + c ∣ x 2 > 1 otherwise d x
=
1
/
|
| / 2
| | -I x
| |--------------------------------------------------------------- for ------------ > 1
| | _____________________________ | a*x3|
| | / 2 ______________________ 2*|c + ----|
| | / x / / a*x3\ | 6 |
| | / -1 + ---------------------- * / polar_lift|c + ----|
| | / / a*x3\ \/ \ 6 /
| | / 2*polar_lift|c + ----|
| |\/ \ 6 /
| < dx
| | 1
| |-------------------------------------------------------------- otherwise
| | ____________________________
| | / 2 ______________________
| | / x / / a*x3\
| | / 1 - ---------------------- * / polar_lift|c + ----|
| | / / a*x3\ \/ \ 6 /
| | / 2*polar_lift|c + ----|
| |\/ \ 6 /
| \
|
/
0
∫ 0 1 { − i x 2 2 polar_lift ( a x 3 6 + c ) − 1 polar_lift ( a x 3 6 + c ) for x 2 2 ∣ a x 3 6 + c ∣ > 1 1 − x 2 2 polar_lift ( a x 3 6 + c ) + 1 polar_lift ( a x 3 6 + c ) otherwise d x \int\limits_{0}^{1} \begin{cases} - \frac{i}{\sqrt{\frac{x^{2}}{2 \operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}} - 1} \sqrt{\operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}}} & \text{for}\: \frac{x^{2}}{2 \left|{\frac{a x_{3}}{6} + c}\right|} > 1 \\\frac{1}{\sqrt{- \frac{x^{2}}{2 \operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}} + 1} \sqrt{\operatorname{polar\_lift}{\left(\frac{a x_{3}}{6} + c \right)}}} & \text{otherwise} \end{cases}\, dx 0 ∫ 1 ⎩ ⎨ ⎧ − 2 polar_lift ( 6 a x 3 + c ) x 2 − 1 polar_lift ( 6 a x 3 + c ) i − 2 polar_lift ( 6 a x 3 + c ) x 2 + 1 polar_lift ( 6 a x 3 + c ) 1 for 2 ∣ 6 a x 3 + c ∣ x 2 > 1 otherwise d x
Integral(Piecewise((-i/(sqrt(-1 + x^2/(2*polar_lift(c + a*x3/6)))*sqrt(polar_lift(c + a*x3/6))), x^2/(2*|c + a*x3/6|) > 1), (1/(sqrt(1 - x^2/(2*polar_lift(c + a*x3/6)))*sqrt(polar_lift(c + a*x3/6))), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.
