Solved example of Integration by substitution

Rewrite the fraction $\frac{2x+1}{\left(x-1\right)^3\left(x^2+4\right)^2}$ in $5$ simpler fractions using partial fraction decomposition

Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by $\left(x-1\right)^3\left(x^2+4\right)^2$

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 decomposed integral equivalent is

The integral of a sum of two or more functions is equal to the sum of their integrals

Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|$, where $b=-1$ and $n=-\frac{1}{66}$

Apply the formula: $\int\frac{n}{\left(x+a\right)^c}dx$$=\frac{-n}{\left(x+a\right)^{\left(c-1\right)}\left(c-1\right)}$, where $a=-1$, $c=3$ and $n=\frac{3}{25}$

Apply the formula: $\frac{a}{bx}$$=\frac{\frac{a}{b}}{x}$, where $a=-\frac{3}{25}$, $b=2$ and $x=\left(x-1\right)^{2}$

Apply the formula: $\int\frac{n}{\left(x+a\right)^c}dx$$=\frac{-n}{\left(x+a\right)^{\left(c-1\right)}\left(c-1\right)}$, where $a=-1$, $c=2$ and $n=-\frac{13}{142}$

Split the fraction $\frac{-243.5608x+-961.7216494845361}{\left(x^2+4\right)^2}$ in two terms with same denominator

Split the fraction $\frac{270.0557x+981.1546391752578}{x^2+4}$ in two terms with same denominator

The integral of a sum of two or more functions is equal to the sum of their integrals

Taking the constant out of the integral

Taking the constant out of the integral

Take the constant out of the integral

Factor the integral's denominator by $4$

Solve the integral $\int\frac{x}{\left(x^2+4\right)^2}dx$ applying u-substitution. Let $u$ and $du$ be

Isolate $dx$ in the previous equation

Substituting $u$ and $dx$ in the integral and simplify

Take the constant out of the integral

Solve the integral applying the substitution $v^2=\frac{1}{4}x^2$

Multiply $2$ times $245.28865979381445$

Solve the integral applying the formula $\displaystyle\int\frac{x'}{x^2+a^2}dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)$

Substitute $v$ back for it's value, $\frac{1}{2}x$

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

Substitute $u$ back for it's value, $x^2+4$

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

Solve the integral $\int\frac{-961.7216}{\left(x^2+4\right)^2}dx$ by trigonometric substitution using the substitution

Substituting in the original integral, we get

Factor by the greatest common divisor $4$

The power of a product is equal to the product of it's factors raised to the same power

Apply the formula: $\frac{ab}{cf}$$=\frac{a}{c}\cdot\frac{b}{f}$, where $a=-1923.4433$, $b=\sec\left(\theta\right)^2$, $c=16$ and $f=\left(\tan\left(\theta\right)^2+1\right)^2$

Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$

Simplifying the fraction by $\sec\left(\theta\right)$

Subtract the values $2$ and $-4$

Apply the formula: $\sec\left(x\right)^n$$=\cos\left(x\right)^{-n}$, where $x=\theta$ and $n=-2$

Take the constant out of the integral

Apply the formula: $\int\cos\left(x\right)^2dx$$=\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)$, where $x=\theta$

Expressing the result of the integral in terms of the original variable

Using the sine double-angle identity

Solve the integral $\int\frac{x}{x^2+4}dx$ applying u-substitution. Let $u$ and $du$ be

Isolate $dx$ in the previous equation

Substituting $u$ and $dx$ in the integral and simplify

Expressing the result of the integral in terms of the original variable

Take the constant out of the integral

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

Substitute $u$ back for it's value, $x^2+4$

As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration