Question(b) $\int \frac{5}{(x+2)\left(x^{2}+1\right)} d x$.

Studdy Solution
Substitute the values of $A$, $B$, and $C$ back into the partial fractions and integrate each term separately:
$\int \frac{5}{(x+2)(x^2+1)} \, dx = \int \frac{1}{x+2} \, dx + \int \frac{-x + 2}{x^2+1} \, dx$
Integrate each term:
1. $\int \frac{1}{x+2} \, dx = \ln|x+2| + C_1$
2. $\int \frac{-x}{x^2+1} \, dx = -\frac{1}{2} \ln|x^2+1| + C_2$ (using substitution $u = x^2+1$)
3. $\int \frac{2}{x^2+1} \, dx = 2 \tan^{-1}(x) + C_3$
Combine the results:
$\int \frac{5}{(x+2)(x^2+1)} \, dx = \ln|x+2| - \frac{1}{2} \ln|x^2+1| + 2 \tan^{-1}(x) + C$
where $C = C_1 + C_2 + C_3$.
The integral evaluates to:
$\ln|x+2| - \frac{1}{2} \ln|x^2+1| + 2 \tan^{-1}(x) + C$