Evaluate the Integral

Problem

(∫_^)(ln(√(,x))*d(x))

Solution

1. Simplify the integrand using the properties of logarithms. Since √(,x)=x(1/2) we can move the exponent to the front of the natural log.

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

2. Factor out the constant coefficient from the integral.

1/2*(∫_^)(ln(x)*d(x))

3. Apply integration by parts to the remaining integral (∫_^)(ln(x)*d(x)) Let u=ln(x) and d(v)=d(x)

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

v=x

4. Substitute into the integration by parts formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

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

5. Simplify the integral of the constant.

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

(∫_^)(ln(x)*d(x))=x*ln(x)−x

6. Combine the result with the constant factor 1/2 and add the constant of integration C

1/2*(x*ln(x)−x)+C

Final Answer

(∫_^)(ln(√(,x))*d(x))=1/2*x*ln(x)−1/2*x+C

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