Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Integrate by substitution
- Integrate by partial fractions
- Integrate by parts
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
- FOIL Method
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Combining like terms $-x^2$ and $8x^2$
Rewrite the fraction $\frac{7x^2-9x+2}{\left(x^2+1\right)\left(x-3\right)^2}$ in $3$ simpler fractions using partial fraction decomposition
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 $\left(x^2+1\right)\left(x-3\right)^2$
Multiplying polynomials
Simplifying
Assigning values to $x$ we obtain the following system of equations
Proceed to solve the system of linear equations
Rewrite as a coefficient matrix
Reducing the original matrix to a identity matrix using Gaussian Elimination
The integral of $\frac{7x^2-9x+2}{\left(x^2+1\right)\left(x-3\right)^2}$ in decomposed fraction equals
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
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
Substituting $u$ and $dx$ in the integral and simplify
The integral $\int\frac{-\frac{51}{50}x+\frac{7}{50}}{x^2+1}dx$ results in: $-\frac{51}{50}\int\frac{x}{x^2+1}dx+\frac{7}{50}\arctan\left(x\right)$
The integral $\frac{1}{5}\int\frac{19}{u^2}du$ results in: $\frac{-19}{5\left(x-3\right)}$
Gather the results of all integrals
We can solve the integral $\int\frac{x}{x^2+1}dx$ 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 $x^2+1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
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
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Multiply $-\frac{51}{50}$ times $\frac{1}{2}$
The integral $-\frac{51}{100}\int\frac{1}{u}du$ results in: $-\frac{51}{100}\ln\left(x^2+1\right)$
The integral $\int\frac{51}{50\left(x-3\right)}dx$ results in: $\frac{51}{50}\ln\left(x-3\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$