Question

First make a substitution and then use integration by parts to evaluate the integral.

Answer 1

The integral is:

`\[ \int x \ln(x) \,dx = (x^2)/(2) \ln(x) - (x^2)/(4) + C \]`

Step-by-step explanation:

To solve the given integral, let's first make the substitution
`\( u = \ln(x) \). This leads to \( du = (1)/(x) \,dx \)`. Now, the integral becomes
`\( \int x \ln(x) \,dx = \int u \,du \). Integrating \( u \) with respect to \( u \) gives \( (u^2)/(2) \).` Substituting back in terms of
`\( x \), we have \( (\ln^2(x))/(2) \).`

Next, use integration by parts with
`\( u = \ln(x) \) and \( dv = x \,dx \). Find \( du \) and \( v \)` accordingly. Applying the integration by parts formula
`\( \int u \,dv = uv - \int v \,du \),` we get the expression
`\( (x^2)/(2) \ln(x) - (x^2)/(4) \).`

Therefore, the final answer is
`\( (x^2)/(2) \ln(x) - (x^2)/(4) + C \), where \( C \)` is the constant of integration. This result combines the outcomes of the substitution and integration by parts, providing a concise expression for the original integral.