Solved example of gaussian elimination
Rewrite the fraction $\frac{1}{x\left(x+1\right)}$ 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 $x\left(x+1\right)$
Multiplying polynomials
Simplifying
Expand the polynomial
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 $\frac{1}{x\left(x+1\right)}$ in decomposed fraction equals
Expand the integral $\int\left(\frac{1}{x}+\frac{-1}{x+1}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
The integral $\int\frac{1}{x}dx$ results in: $\ln\left(x\right)$
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left(x+b\right)+C$, where $b=1$ and $n=1$
The integral $\int\frac{-1}{x+1}dx$ results in: $-\ln\left(x+1\right)$
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 $C$
Access detailed step by step solutions to thousands of problems, growing every day!