Sr Examen
Integral de (sin^3(x))/(1+cos^2(x))dx dx
Solución
Ha introducido
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1 / | | 3 | sin (x) | ----------- dx | 2 | 1 + cos (x) | / 0
$$\int\limits_{0}^{1} \frac{\sin^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 1}\, dx$$
Integral(sin(x)^3/(1 + cos(x)^2), (x, 0, 1))
Respuesta (Indefinida)
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/ /x pi\ \ / /x pi\ \ / /x pi\ \ / /x pi\ \ / | |- - --| | | |- - --| | | |- - --| | | |- - --| | | | |2 2 | / ___ /x\\| | |2 2 | / ___ /x\\| 2/x\ | |2 2 | / ___ /x\\| 2/x\ | |2 2 | / ___ /x\\| | 3 2*|pi*floor|------| + atan|1 + \/ 2 *tan|-||| 2*|pi*floor|------| + atan|-1 + \/ 2 *tan|-||| 2*tan |-|*|pi*floor|------| + atan|1 + \/ 2 *tan|-||| 2*tan |-|*|pi*floor|------| + atan|-1 + \/ 2 *tan|-||| | sin (x) 2 \ \ pi / \ \2/// \ \ pi / \ \2/// \2/ \ \ pi / \ \2/// \2/ \ \ pi / \ \2/// | ----------- dx = C + ----------- - --------------------------------------------- + ---------------------------------------------- - ----------------------------------------------------- + ------------------------------------------------------ | 2 2/x\ 2/x\ 2/x\ 2/x\ 2/x\ | 1 + cos (x) 1 + tan |-| 1 + tan |-| 1 + tan |-| 1 + tan |-| 1 + tan |-| | \2/ \2/ \2/ \2/ \2/ /
$$\int \frac{\sin^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)} + 1}\, dx = C + \frac{2 \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \left(\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
Gráfica
Respuesta
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/ / ___ \\ / / ___ \\ 2 / / ___ \\ 2 / / ___ \\
2 2*\-pi + atan\1 + \/ 2 *tan(1/2)// 2*\-pi - atan\1 - \/ 2 *tan(1/2)// 2*tan (1/2)*\-pi + atan\1 + \/ 2 *tan(1/2)// 2*tan (1/2)*\-pi - atan\1 - \/ 2 *tan(1/2)//
-2 + pi + ------------- - ---------------------------------- + ---------------------------------- - -------------------------------------------- + --------------------------------------------
2 2 2 2 2
1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2)
$$\frac{2 \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - 2 + \frac{2 \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \pi - \frac{2 \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{\tan^{2}{\left(\frac{1}{2} \right)} + 1}$$
=
=
/ / ___ \\ / / ___ \\ 2 / / ___ \\ 2 / / ___ \\
2 2*\-pi + atan\1 + \/ 2 *tan(1/2)// 2*\-pi - atan\1 - \/ 2 *tan(1/2)// 2*tan (1/2)*\-pi + atan\1 + \/ 2 *tan(1/2)// 2*tan (1/2)*\-pi - atan\1 - \/ 2 *tan(1/2)//
-2 + pi + ------------- - ---------------------------------- + ---------------------------------- - -------------------------------------------- + --------------------------------------------
2 2 2 2 2
1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2) 1 + tan (1/2)
$$\frac{2 \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - 2 + \frac{2 \left(- \pi - \operatorname{atan}{\left(- \sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right) \tan^{2}{\left(\frac{1}{2} \right)}}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2}{\tan^{2}{\left(\frac{1}{2} \right)} + 1} + \pi - \frac{2 \left(- \pi + \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{1}{2} \right)} + 1 \right)}\right)}{\tan^{2}{\left(\frac{1}{2} \right)} + 1}$$
-2 + pi + 2/(1 + tan(1/2)^2) - 2*(-pi + atan(1 + sqrt(2)*tan(1/2)))/(1 + tan(1/2)^2) + 2*(-pi - atan(1 - sqrt(2)*tan(1/2)))/(1 + tan(1/2)^2) - 2*tan(1/2)^2*(-pi + atan(1 + sqrt(2)*tan(1/2)))/(1 + tan(1/2)^2) + 2*tan(1/2)^2*(-pi - atan(1 - sqrt(2)*tan(1/2)))/(1 + tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.