Evaluate the Integral

Problem

(∫_^)(ln(√(3,x))*d(x))

Solution

1. Simplify the integrand using the properties of logarithms, specifically ln(ab)=b*ln(a)

√(3,x)=x(1/3)

ln(x(1/3))=1/3*ln(x)

2. Factor out the constant from the integral.

(∫_^)(1/3*ln(x)*d(x))=1/3*(∫_^)(ln(x)*d(x))

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

u=ln(x)⇒d(u)=1/x*d(x)

d(v)=d(x)⇒v=x

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

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

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

(∫_^)(ln(x)*d(x))=x*ln(x)−x

5. Multiply the result by the constant factor 1/3 and add the constant of integration C

1/3*(x*ln(x)−x)+C

(x*ln(x))/3−x/3+C

Final Answer

(∫_^)(ln(√(3,x))*d(x))=(x*ln(x)−x)/3+C

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