I am a student, and I was assigned this integral as homework:
$\int_0^{2\sqrt{2}} \frac{x}{\sqrt{1+x^2}} \, dx$
I am a student, and I was assigned this integral as homework:
$\int_0^{2\sqrt{2}} \frac{x}{\sqrt{1+x^2}} \, dx$
There is no need for substitutions $\int_0^{2\sqrt{2}} \frac{(1+x^2)'}{2\sqrt{1+x^2}}dx=\int_0^{2\sqrt{2}} ({\sqrt{1+x^2}})'dx=\sqrt{1+x^2}|_0^{2\sqrt{2}}=2$
Hints:
You can use "$u$-substitution" to simplify matters:
Let $u = 1 + x^2$. Then $du = \;\;?\;dx,\;\;$ and note that $\int_0^{2\sqrt{2}} \frac{x}{\sqrt{1+x^2}}dx = \frac{1}{2}\int_0^{2\sqrt{2}} \frac{2x\; dx}{\sqrt{(1+x^2)}}$
Integrate with respect to $u$, with the new bounds of integration being:
from $\;u\;$ evaluated at $\;x = 0: \; u = 1 + 0^2 = 1\;$
to $\;u\;$ evaluated at $\;x = 2\sqrt{2}:\;1 + (2\sqrt{2})^2 = 1 + 8 = 9$.
This is an easy and elementary definite integral. First of all consider $\int\frac{x}{\sqrt{1+x^2}} \, dx$. You can solve it by taking $u=1+x^2$ inside the radical and use it as a good substitution. Then $u=1+x^2\longrightarrow du=2x \, dx\longrightarrow x \, dx=\frac{du}{2}$ so the indifinte integral would be $\int\frac{du/2}{\sqrt{u}}.$ I think you can handle it.