To integrate the function ln x², we can use integration by parts. The integration by parts formula is:
∫u dv = uv – ∫v du
First, we need to select our u and dv. Let’s choose:
- u = ln x² (thus, we will differentiate this)
- dv = dx (and we will integrate this)
Next, we need to compute du and v:
- du = (1/x²) * 2x dx = (2/x) dx
- v = x
Now, we can apply the integration by parts formula:
∫ln x² dx = x ln x² – ∫x * (2/x) dx
This simplifies to:
∫ln x² dx = x ln x² – 2∫dx
Integrating 2∫dx gives us:
∫ln x² dx = x ln x² – 2x + C
where C is the constant of integration.
To summarize, the integral of ln x² is:
∫ln x² dx = x ln x² – 2x + C