x.lnx2 integral | What is integral of x.lnx^2? | ∫x.lnx2.dx=?
Integral of x * ln(x²) | What is the Integral of x * ln(x²)?
In calculus, integrals play a significant role in finding areas under curves, among other applications. In this article, we will explore the process of solving the integral of the function ( x ln(x^2) ).
Understanding the Function
The function we want to integrate is ( x ln(x^2) ). First, let’s simplify it using properties of logarithms:
$ln(x^2) = 2 ln(x)$
This transforms our integral to:
$int x ln(x^2) , dx = int x (2 ln(x)) , dx = 2 int x ln(x) , dx$
Now we focus on finding the integral of ( x ln(x) ).
Integration by Parts
To calculate ( int x ln(x) , dx ), we will use the method of integration by parts. The formula for integration by parts is:
$int u , dv = uv – int v , du$
Let’s set:
( u = ln(x) ) –> which implies ( du = frac{1}{x} , dx )
( dv = x , dx ) –> which gives us ( v = frac{x^2}{2} )
Now we can substitute into the integration by parts formula:
$int x ln(x) , dx = ln(x) cdot frac{x^2}{2} – int frac{x^2}{2} cdot frac{1}{x} , dx$
This simplifies to:
$int x ln(x) , dx = frac{x^2}{2} ln(x) – int frac{x^2}{2x} , dx = frac{x^2}{2} ln(x) – int frac{x}{2} , dx$
Next, we compute ( int frac{x}{2} , dx ):
$int frac{x}{2} , dx = frac{1}{2} cdot frac{x^2}{2} = frac{x^2}{4}$
Compiling the Results
Putting all of the pieces together, we have:
$int x ln(x) , dx = frac{x^2}{2} ln(x) – frac{x^2}{4} + C$
where ( C ) is the constant of integration.
Final Result
Now, we can return to our original integral:
$int x ln(x^2) , dx = 2 int x ln(x) , dx = 2 left( frac{x^2}{2} ln(x) – frac{x^2}{4} right) + C$
This simplifies to:
$int x ln(x^2) , dx = x^2 ln(x) – frac{x^2}{2} + C$
Conclusion
Thus, the integral of the function ( x ln(x^2) ) is given by:
$int x ln(x^2) , dx = x^2 ln(x) – frac{x^2}{2} + C$
This result showcases the beauty of calculus and the power of integration by parts, allowing us to tackle more complex functions systematically. Understanding these techniques extends our ability to solve a vast array of calculus problems efficiently.
