Evaluate the Integral

Problem

(∫_^)(2*ln(x)*d(x))

Solution

1. Factor out the constant from the integral to simplify the expression.

(∫_^)(2*ln(x)*d(x))=2*(∫_^)(ln(x)*d(x))

2. Apply integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=ln(x) and d(v)=d(x)

u=ln(x)

d(v)=d(x)

3. Differentiate u and integrate d(v) to find d(u) and v

d(u)=1/x*d(x)

v=x

4. Substitute these values into the integration by parts formula.

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

5. Simplify the integrand inside the remaining integral.

2*(x*ln(x)−(∫_^)(1*d(x)))

6. Evaluate the integral of 1 and distribute the constant.

2*(x*ln(x)−x)+C

7. Factor out the common term x to reach the final form.

2*x*ln(x)−2*x+C

Final Answer

(∫_^)(2*ln(x)*d(x))=2*x*ln(x)−2*x+C

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