Question

(a) Evaluate∫[

𝒙/(𝒙^2+𝟏)^(1/2)𝒅𝒙.

(b) Use MATLAB to generate some typical integral curves of 𝑓(𝑥) =

𝒙/(𝒙^2+𝟏)^(1/2)𝒅𝒙over the interval (−5,5).

Answer 1

Solution:

(a):

$I=\int \dfrac{x}{\sqrt{x^2+1}}dx$

$I=\int \dfrac{x}{\sqrt{x^2+1}}dx$

Put $x^2+1=t$

$\Rightarrow 2xdx=dt \\\Rightarrow xdx=\dfrac{dt}{2}$

So, $I=\dfrac{1}{2}\int \dfrac{1}{\sqrt{t}}dt$

$I=\dfrac{1}{2}\int \dfrac{1}{\sqrt{t}}dt$

$I=\dfrac{1}{2}\int t^{-1/2}dt \\=\dfrac{1}{2}.\dfrac{t^{1/2}}{1/2}+c \\=\sqrt t+c \\=\sqrt{x^2+1}+c$

$I=\dfrac{1}{2}\int t^{-1/2}dt \\=\dfrac{1}{2}.\dfrac{t^{1/2}}{1/2}+c \\=\sqrt t+c \\=\sqrt{x^2+1}+c$$I=\dfrac{1}{2}\int t^{-1/2}dt \\=\dfrac{1}{2}.\dfrac{t^{1/2}}{1/2}+c \\=\sqrt t+c \\=\sqrt{x^2+1}+c$

(b):

The graph using matlab from (-5,5) is as follows: