Sr Examen
Integral de 1/x*sqrt(x^2+1) dx
Solución
Ha introducido
[src]
1 / | | ________ | / 2 | \/ x + 1 | ----------- dx | x | / 2 a
$$\int\limits_{a^{2}}^{1} \frac{\sqrt{x^{2} + 1}}{x}\, dx$$
Integral(sqrt(x^2 + 1)/x, (x, a^2, 1))
Respuesta (Indefinida)
[src]
/
|
| ________
| / 2
| \/ x + 1 /1\ x 1
| ----------- dx = C - asinh|-| + ------------- + ---------------
| x \x/ ________ ________
| / 1 / 1
/ / 1 + -- x* / 1 + --
/ 2 / 2
\/ x \/ x
$$\int \frac{\sqrt{x^{2} + 1}}{x}\, dx = C + \frac{x}{\sqrt{1 + \frac{1}{x^{2}}}} - \operatorname{asinh}{\left(\frac{1}{x} \right)} + \frac{1}{x \sqrt{1 + \frac{1}{x^{2}}}}$$
Respuesta
[src]
2
___ / ___\ 1 a /1 \
\/ 2 - log\1 + \/ 2 / - ---------------- - ------------- + asinh|--|
________ ________ | 2|
2 / 1 / 1 \a /
a * / 1 + -- / 1 + --
/ 4 / 4
\/ a \/ a
$$- \frac{a^{2}}{\sqrt{1 + \frac{1}{a^{4}}}} + \operatorname{asinh}{\left(\frac{1}{a^{2}} \right)} - \log{\left(1 + \sqrt{2} \right)} + \sqrt{2} - \frac{1}{a^{2} \sqrt{1 + \frac{1}{a^{4}}}}$$
=
=
2
___ / ___\ 1 a /1 \
\/ 2 - log\1 + \/ 2 / - ---------------- - ------------- + asinh|--|
________ ________ | 2|
2 / 1 / 1 \a /
a * / 1 + -- / 1 + --
/ 4 / 4
\/ a \/ a
$$- \frac{a^{2}}{\sqrt{1 + \frac{1}{a^{4}}}} + \operatorname{asinh}{\left(\frac{1}{a^{2}} \right)} - \log{\left(1 + \sqrt{2} \right)} + \sqrt{2} - \frac{1}{a^{2} \sqrt{1 + \frac{1}{a^{4}}}}$$
sqrt(2) - log(1 + sqrt(2)) - 1/(a^2*sqrt(1 + a^(-4))) - a^2/sqrt(1 + a^(-4)) + asinh(a^(-2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.