Sr Examen
Integral de sqrt(1+sqrt(1+x^2)) dx
Solución
Ha introducido
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1 / | | _________________ | / ________ | / / 2 | \/ 1 + \/ 1 + x dx | / 0
$$\int\limits_{0}^{1} \sqrt{\sqrt{x^{2} + 1} + 1}\, dx$$
Integral(sqrt(1 + sqrt(1 + x^2)), (x, 0, 1))
Respuesta (Indefinida)
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| _________________ ________
| / ________ ___ 3 ___ ___ / 2
| / / 2 \/ 2 *x *Gamma(-1/4)*Gamma(1/4) 3*x*\/ 2 *Gamma(-1/4)*Gamma(1/4) 3*x*\/ 2 *\/ 1 + x *Gamma(-1/4)*Gamma(1/4)
| \/ 1 + \/ 1 + x dx = C - --------------------------------------------------------------------- - --------------------------------------------------------------------- - ---------------------------------------------------------------------
| _________________ _________________ _________________ _________________ _________________ _________________
/ / ________ ________ / ________ / ________ ________ / ________ / ________ ________ / ________
/ / 2 / 2 / / 2 / / 2 / 2 / / 2 / / 2 / 2 / / 2
12*pi*\/ 1 + \/ 1 + x + 12*pi*\/ 1 + x *\/ 1 + \/ 1 + x 12*pi*\/ 1 + \/ 1 + x + 12*pi*\/ 1 + x *\/ 1 + \/ 1 + x 12*pi*\/ 1 + \/ 1 + x + 12*pi*\/ 1 + x *\/ 1 + \/ 1 + x
$$\int \sqrt{\sqrt{x^{2} + 1} + 1}\, dx = C - \frac{\sqrt{2} x^{3} \Gamma\left(- \frac{1}{4}\right) \Gamma\left(\frac{1}{4}\right)}{12 \pi \sqrt{x^{2} + 1} \sqrt{\sqrt{x^{2} + 1} + 1} + 12 \pi \sqrt{\sqrt{x^{2} + 1} + 1}} - \frac{3 \sqrt{2} x \sqrt{x^{2} + 1} \Gamma\left(- \frac{1}{4}\right) \Gamma\left(\frac{1}{4}\right)}{12 \pi \sqrt{x^{2} + 1} \sqrt{\sqrt{x^{2} + 1} + 1} + 12 \pi \sqrt{\sqrt{x^{2} + 1} + 1}} - \frac{3 \sqrt{2} x \Gamma\left(- \frac{1}{4}\right) \Gamma\left(\frac{1}{4}\right)}{12 \pi \sqrt{x^{2} + 1} \sqrt{\sqrt{x^{2} + 1} + 1} + 12 \pi \sqrt{\sqrt{x^{2} + 1} + 1}}$$
Gráfica
Respuesta
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6*Gamma(-1/4)*Gamma(1/4) 4*\/ 2 *Gamma(-1/4)*Gamma(1/4)
- ------------------------------------------------- - -------------------------------------------------
___________ ___________ ___________ ___________
/ ___ ___ / ___ / ___ ___ / ___
12*pi*\/ 1 + \/ 2 + 12*pi*\/ 2 *\/ 1 + \/ 2 12*pi*\/ 1 + \/ 2 + 12*pi*\/ 2 *\/ 1 + \/ 2
$$- \frac{4 \sqrt{2} \Gamma\left(- \frac{1}{4}\right) \Gamma\left(\frac{1}{4}\right)}{12 \pi \sqrt{1 + \sqrt{2}} + 12 \sqrt{2} \pi \sqrt{1 + \sqrt{2}}} - \frac{6 \Gamma\left(- \frac{1}{4}\right) \Gamma\left(\frac{1}{4}\right)}{12 \pi \sqrt{1 + \sqrt{2}} + 12 \sqrt{2} \pi \sqrt{1 + \sqrt{2}}}$$
=
=
___
6*Gamma(-1/4)*Gamma(1/4) 4*\/ 2 *Gamma(-1/4)*Gamma(1/4)
- ------------------------------------------------- - -------------------------------------------------
___________ ___________ ___________ ___________
/ ___ ___ / ___ / ___ ___ / ___
12*pi*\/ 1 + \/ 2 + 12*pi*\/ 2 *\/ 1 + \/ 2 12*pi*\/ 1 + \/ 2 + 12*pi*\/ 2 *\/ 1 + \/ 2
$$- \frac{4 \sqrt{2} \Gamma\left(- \frac{1}{4}\right) \Gamma\left(\frac{1}{4}\right)}{12 \pi \sqrt{1 + \sqrt{2}} + 12 \sqrt{2} \pi \sqrt{1 + \sqrt{2}}} - \frac{6 \Gamma\left(- \frac{1}{4}\right) \Gamma\left(\frac{1}{4}\right)}{12 \pi \sqrt{1 + \sqrt{2}} + 12 \sqrt{2} \pi \sqrt{1 + \sqrt{2}}}$$
-6*gamma(-1/4)*gamma(1/4)/(12*pi*sqrt(1 + sqrt(2)) + 12*pi*sqrt(2)*sqrt(1 + sqrt(2))) - 4*sqrt(2)*gamma(-1/4)*gamma(1/4)/(12*pi*sqrt(1 + sqrt(2)) + 12*pi*sqrt(2)*sqrt(1 + sqrt(2)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.