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 $s\left(s^2+10s+10\right)$
We can solve the integral $\int\frac{s}{-15+\left(s+5\right)^2}ds$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $s+5$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
$u=s+5$
Intermediate steps
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Now, in order to rewrite $ds$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
$du=ds$
Intermediate steps
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Rewriting $s$ in terms of $u$
$s=u-5$
Intermediate steps
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Substituting $u$, $ds$ and $s$ in the integral and simplify
$\ln\left(s\right)-\int\frac{u-5}{-15+u^2}du$
Intermediate steps
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The integral $-\int\frac{u-5}{-15+u^2}du$ results in: $-\frac{1}{2}\ln\left(-15+\left(s+5\right)^2\right)-\frac{5}{2\sqrt{15}}\ln\left(\sqrt{15}+s+5\right)+\frac{5}{2\sqrt{15}}\ln\left(s+5-\sqrt{15}\right)$
Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more
The partial fraction decomposition or partial fraction expansion of a rational function is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.