Here, we show you a step-by-step solved example of gaussian elimination. This solution was automatically generated by our smart calculator:
Take the constant $\frac{1}{2}$ out of the integral
Rewrite the fraction $\frac{1}{x^2\left(x-1\right)}$ in $3$ simpler fractions using partial fraction decomposition
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 $x^2\left(x-1\right)$
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 $\frac{1}{x^2\left(x-1\right)}$ in decomposed fractions equals
Rewrite the fraction $\frac{1}{x^2\left(x-1\right)}$ in $3$ simpler fractions using partial fraction decomposition
Expand the integral $\int\left(\frac{-1}{x^2}+\frac{1}{x-1}+\frac{-1}{x}\right)dx$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately
We can solve the integral $\int\frac{1}{x-1}dx$ 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 $x-1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Differentiate both sides of the equation $u=x-1$
Find the derivative
The derivative of a sum of two or more functions is the sum of the derivatives of each function
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
Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
Multiply the fraction and term in $-\left(\frac{1}{2}\right)\int x^{-2}dx$
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $-2$
Simplify the expression
The integral $\frac{1}{2}\int\frac{-1}{x^2}dx$ results in: $\frac{1}{2x}$
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Replace $u$ with the value that we assigned to it in the beginning: $x-1$
The integral $\frac{1}{2}\int\frac{1}{u}du$ results in: $\frac{1}{2}\ln\left|x-1\right|$
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Multiply the fraction and term in $-\left(\frac{1}{2}\right)\ln\left|x\right|$
The integral $\frac{1}{2}\int\frac{-1}{x}dx$ results in: $-\frac{1}{2}\ln\left|x\right|$
Gather the results of all integrals
Add the initial limits of integration
Replace the integral's limit by a finite value
Evaluate the definite integral
The natural log of infinity is equal to infinity, $\lim_{x\to\infty}\ln(x)=\infty$
Any expression multiplied by infinity tends to infinity, in other words: $\infty\cdot(\pm n)=\pm\infty$, if $n\neq0$
Any expression multiplied by infinity tends to infinity, in other words: $\infty\cdot(\pm n)=\pm\infty$, if $n\neq0$
Infinity plus any algebraic expression is equal to infinity
The natural log of infinity is equal to infinity, $\lim_{x\to\infty}\ln(x)=\infty$
Any expression multiplied by infinity tends to infinity, in other words: $\infty\cdot(\pm n)=\pm\infty$, if $n\neq0$
Any expression divided by infinity is equal to zero
Infinity minus infinity is an indeterminate form
Evaluate the resulting limits of the integral
When the limits of the integral do not exist, it is said that the integral is divergent
Access detailed step by step solutions to thousands of problems, growing every day!
Most popular problems solved with this calculator: