Final Answer
$\ln\left(s\right)-\frac{5}{\sqrt{15}}\ln\left(\frac{8.872983+s}{\sqrt{-15+\left(s+5\right)^2}}\right)+\ln\left(\frac{\sqrt{15}}{\sqrt{-15+\left(s+5\right)^2}}\right)+C_0$
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Step-by-step Solution
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1
Rewrite the fraction $\frac{10s+10}{s\left(s^2+10s+10\right)}$ in $2$ simpler fractions using partial fraction decomposition
$\frac{10s+10}{s\left(s^2+10s+10\right)}=\frac{A}{s}+\frac{Bs+C}{s^2+10s+10}$
2
Find the values for the unknown coefficients: $A, B, C$. The first step is to multiply both sides of the equation from the previous step by $s\left(s^2+10s+10\right)$
$10s+10=s\left(s^2+10s+10\right)\left(\frac{A}{s}+\frac{Bs+C}{s^2+10s+10}\right)$
3
Multiplying polynomials
$10s+10=\frac{s\left(s^2+10s+10\right)A}{s}+\frac{s\left(s^2+10s+10\right)\left(Bs+C\right)}{s^2+10s+10}$
$10s+10=\left(s^2+10s+10\right)A+s\left(Bs+C\right)$
5
Assigning values to $s$ we obtain the following system of equations
$\begin{matrix}10=10A&\:\:\:\:\:\:\:(s=0) \\ -90=10A+100B-10C&\:\:\:\:\:\:\:(s=-10) \\ 110=210A+100B+10C&\:\:\:\:\:\:\:(s=10)\end{matrix}$
6
Proceed to solve the system of linear equations
$\begin{matrix}10A & + & 0B & + & 0C & =10 \\ 10A & + & 100B & - & 10C & =-90 \\ 210A & + & 100B & + & 10C & =110\end{matrix}$
7
Rewrite as a coefficient matrix
$\left(\begin{matrix}10 & 0 & 0 & 10 \\ 10 & 100 & -10 & -90 \\ 210 & 100 & 10 & 110\end{matrix}\right)$
8
Reducing the original matrix to a identity matrix using Gaussian Elimination
$\left(\begin{matrix}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 0\end{matrix}\right)$
9
The integral of $\frac{10s+10}{s\left(s^2+10s+10\right)}$ in decomposed fraction equals
$\int\left(\frac{1}{s}+\frac{-s}{s^2+10s+10}\right)ds$
Intermediate steps
10
Simplify the expression inside the integral
$\int\frac{1}{s}ds-\int\frac{s}{s^2+10s+10}ds$
Explain this step further
Intermediate steps
11
The integral $\int\frac{1}{s}ds$ results in: $\ln\left(s\right)$
$\ln\left(s\right)$
Explain this step further
12
Gather the results of all integrals
$\ln\left(s\right)-\int\frac{s}{s^2+10s+10}ds$
Intermediate steps
13
Rewrite the expression $\frac{s}{s^2+10s+10}$ inside the integral in factored form
$\ln\left(s\right)-\int\frac{s}{-15+\left(s+5\right)^2}ds$
Explain this step further
Intermediate steps
14
The integral $-\int\frac{s}{-15+\left(s+5\right)^2}ds$ results in: $\ln\left(\frac{\sqrt{15}}{\sqrt{-15+\left(s+5\right)^2}}\right)-\frac{5}{\sqrt{15}}\ln\left(\frac{s+5}{\sqrt{-15+\left(s+5\right)^2}}+\frac{\sqrt{15}}{\sqrt{-15+\left(s+5\right)^2}}\right)$
$\ln\left(\frac{\sqrt{15}}{\sqrt{-15+\left(s+5\right)^2}}\right)-\frac{5}{\sqrt{15}}\ln\left(\frac{s+5}{\sqrt{-15+\left(s+5\right)^2}}+\frac{\sqrt{15}}{\sqrt{-15+\left(s+5\right)^2}}\right)$
Explain this step further
15
Gather the results of all integrals
$\ln\left(s\right)-\frac{5}{\sqrt{15}}\ln\left(\frac{s+5}{\sqrt{-15+\left(s+5\right)^2}}+\frac{\sqrt{15}}{\sqrt{-15+\left(s+5\right)^2}}\right)+\ln\left(\frac{\sqrt{15}}{\sqrt{-15+\left(s+5\right)^2}}\right)$
16
The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
$L.C.M.=\sqrt{-15+\left(s+5\right)^2}$
Intermediate steps
17
Combine and simplify all terms in the same fraction with common denominator $\sqrt{-15+\left(s+5\right)^2}$
$\ln\left(s\right)-\frac{5}{\sqrt{15}}\ln\left(\frac{8.872983+s}{\sqrt{-15+\left(s+5\right)^2}}\right)+\ln\left(\frac{\sqrt{15}}{\sqrt{-15+\left(s+5\right)^2}}\right)$
Explain this step further
18
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\ln\left(s\right)-\frac{5}{\sqrt{15}}\ln\left(\frac{8.872983+s}{\sqrt{-15+\left(s+5\right)^2}}\right)+\ln\left(\frac{\sqrt{15}}{\sqrt{-15+\left(s+5\right)^2}}\right)+C_0$
Final Answer$\ln\left(s\right)-\frac{5}{\sqrt{15}}\ln\left(\frac{8.872983+s}{\sqrt{-15+\left(s+5\right)^2}}\right)+\ln\left(\frac{\sqrt{15}}{\sqrt{-15+\left(s+5\right)^2}}\right)+C_0$