Evaluate the Integral

Problem

(∫_^)(ln(x2)*d(x))

Solution

1. Simplify the integrand using the power rule for logarithms, ln(ab)=b*ln(a) to make the integration easier.

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

2. Factor out the constant from the integral.

2*(∫_^)(ln(x)*d(x))

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

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

v=x

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

2*(x*ln(x)−(∫_^)(x⋅1/x*d(x)))

5. Simplify the integrand inside the new integral.

2*(x*ln(x)−(∫_^)(1*d(x)))

6. Evaluate the remaining integral.

2*(x*ln(x)−x)+C

7. Distribute the constant to reach the final form.

2*x*ln(x)−2*x+C

Final Answer

(∫_^)(ln(x2)*d(x))=2*x*ln(x)−2*x+C

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