Evaluate the Integral integral of x log of x with respect to x

Problem

(∫_^)(x*ln(x)*d(x))

Solution

1. Identify the method of integration by parts, which uses the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Choose the components for the formula by letting u=ln(x) and d(v)=x*d(x)

3. Differentiate u to find d(u)=1/x*d(x)

4. Integrate d(v) to find v=(x2)/2

5. Substitute these components into the integration by parts formula.

(∫_^)(x*ln(x)*d(x))=(x2)/2*ln(x)−(∫_^)((x2)/2⋅1/x*d(x))

6. Simplify the integrand in the second term.

(∫_^)(x*ln(x)*d(x))=(x2)/2*ln(x)−(∫_^)(x/2*d(x))

7. Evaluate the remaining integral.

(∫_^)(x*ln(x)*d(x))=(x2)/2*ln(x)−(x2)/4+C

Final Answer

(∫_^)(x*ln(x)*d(x))=(x2*ln(x))/2−(x2)/4+C

Want more problems? Check here!