jubateee

2021-12-30

Determine the following indefinite integral.
$\int \frac{2+{x}^{2}}{1+{x}^{2}}dx$

### Answer & Explanation

Foreckije

Step 1
Given: indefinite integral $\int \frac{2+{x}^{2}}{1+{x}^{2}}dx$
To solve the indefinite integral, first simplify the function,
$\frac{2+{x}^{2}}{1+{x}^{2}}=\frac{1+{x}^{2}+1}{1+{x}^{2}}$
$=1+\frac{1}{1+{x}^{2}}$
Step 2
Substitute simplified function in the given integral.
$\int \frac{2+{x}^{2}}{1+{x}^{2}}dx=\int 1+\frac{1}{1+{x}^{2}}dx$
$=\int \frac{1}{1+{x}^{2}}dx+\int 1dx$
$={\mathrm{tan}}^{-1}x+x+c$
Step 3
Check the solution by differentiation
$\frac{d}{dx}\left(\mathrm{arctan}\left(x\right)+x+c\right)=\frac{d}{dx}\left(\mathrm{arctan}\left(x\right)\right)+\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(c\right)$
$=\frac{1}{{x}^{2}+1}+1+0$
$=\frac{1+{x}^{2}+1}{{x}^{2}+1}$
$=\frac{2+{x}^{2}}{{x}^{2}+1}$
Therefore, this matches with the given integral.

redhotdevil13l3

$\int \frac{2+{x}^{2}}{1+{x}^{2}}dx$
Rewrite the expression
$\int \frac{1+1+{x}^{2}}{1+{x}^{2}}dx$
Separate the fraction
$\int \frac{1}{1+{x}^{2}}+\frac{1+{x}^{2}}{1+{x}^{2}}dx$
Use properties of integrals
$\int \frac{1}{1+{x}^{2}}dx+\int \frac{1+{x}^{2}}{1={x}^{2}}dx$
Evaluate the integrals
$\mathrm{arctan}\left(x\right)+x$
Solution
$\mathrm{arctan}\left(x\right)+x+C$

karton

$\begin{array}{}\frac{{x}^{2}+2}{{x}^{2}+1}dx\\ =\int \left(\frac{{x}^{2}+1}{{x}^{2}+1}+\frac{1}{{x}^{2}+1}\right)dx\\ =\int \left(\frac{1}{{x}^{2}+1}+1\right)dx\\ =\int \frac{1}{{x}^{2}+1}dx+\int 1dx\\ \int \frac{1}{{x}^{2}+1}dx\\ =\mathrm{arctan}\left(x\right)\\ \int 1dx\\ =x\\ \int \frac{1}{{x}^{2}+1}dx+\int 1dx\\ =\mathrm{arctan}\left(x\right)+x\\ \int \frac{{x}^{2}+2}{{x}^{2}+1}\\ =\mathrm{arctan}\left(x\right)+x+C\end{array}$