What is the integral of ∫x ln x?

The integral of the function ∫x ln x can be solved using integration by parts. This method is useful when dealing with the product of two functions, in this case, x and ln x.

Steps to Solve the Integral

We will apply the integration by parts formula:

• Formula: ∫u dv = uv - ∫v du

Choosing u and dv

For our integral, we can set:

• u = ln x (which means du = (1/x) dx)
• dv = x dx (which means v = (1/2)x²)

Applying the Formula

Now, we substitute these into the integration by parts formula:

• ∫x ln x dx = (ln x)(1/2)x² - ∫(1/2)x²(1/x) dx
• = (1/2)x² ln x - (1/2)∫x dx

Calculating the Remaining Integral

The remaining integral ∫x dx is straightforward:

• ∫x dx = (1/2)x²

Final Result

Putting it all together, we have:

• ∫x ln x dx = (1/2)x² ln x - (1/2)(1/2)x² + C
• = (1/2)x² ln x - (1/4)x² + C

Thus, the integral of ∫x ln x is:

(1/2)x² ln x - (1/4)x² + C