Find the values for the unknown coefficients: $A, B, C, D$. The first step is to multiply both sides of the equation from the previous step by $\left(x^2+1\right)\left(x-3\right)^2$
Expand the integral $\int\left(\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}+\frac{19}{5\left(x-3\right)^2}+\frac{51}{50\left(x-3\right)}\right)dx$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately
We can solve the integral $\int\frac{19}{5\left(x-3\right)^2}dx$ 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 $x-3$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
$u=x-3$
Intermediate steps
13
Now, in order to rewrite $dx$ 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
$du=dx$
Intermediate steps
14
Substituting $u$ and $dx$ in the integral and simplify
We can solve the integral $-\frac{51}{50}\int\frac{x}{x^2+1}dx$ by applying integration method of trigonometric substitution using the substitution
$x=\tan\left(\theta \right)$
Intermediate steps
19
Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above
We can solve the integral $\int\frac{51}{50\left(x-3\right)}dx$ 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 $x-3$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
$u=x-3$
Intermediate steps
24
Now, in order to rewrite $dx$ 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
$du=dx$
Intermediate steps
25
Substituting $u$ and $dx$ in the integral and simplify
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.