Here, we show you a step-by-step solved example of matrices. This solution was automatically generated by our smart calculator:
Rewrite the fraction $\frac{32x-20}{\left(x-1\right)\left(5x-3\right)}$ in $2$ simpler fractions using partial fraction decomposition
Find the values for the unknown coefficients: $A, B$. The first step is to multiply both sides of the equation from the previous step by $\left(x-1\right)\left(5x-3\right)$
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 $\frac{32x-20}{\left(x-1\right)\left(5x-3\right)}$ in decomposed fractions equals
Rewrite the fraction $\frac{32x-20}{\left(x-1\right)\left(5x-3\right)}$ in $2$ simpler fractions using partial fraction decomposition
Expand the integral $\int\left(\frac{6}{x-1}+\frac{2}{5x-3}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
Apply the formula: $\int\frac{n}{x+b}dx$$=nsign\left(x\right)\ln\left(x+b\right)+C$, where $b=-1$ and $n=6$
The integral $\int\frac{6}{x-1}dx$ results in: $6\ln\left(x-1\right)$
Apply the formula: $\int\frac{n}{a+b}dx$$=n\int\frac{1}{a+b}dx$, where $a=-3$, $b=5x$ and $n=2$
Apply the formula: $\int\frac{n}{ax+b}dx$$=\frac{n}{a}\ln\left(ax+b\right)+C$, where $a=5$, $b=-3$ and $n=1$
Apply the formula: $\frac{a}{b}c$$=\frac{ca}{b}$, where $a=1$, $b=5$, $c=2$, $a/b=\frac{1}{5}$ and $ca/b=2\cdot \left(\frac{1}{5}\right)\ln\left(5x-3\right)$
The integral $\int\frac{2}{5x-3}dx$ results in: $\frac{2}{5}\ln\left(5x-3\right)$
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 $C$
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