Question

Evaluate the integral. (Use C for the constant of integration.) \int \ln (\sqrt x)dx

Integrals
ANSWERED
asked 2021-05-02
Evaluate the integral. (Use C for the constant of integration.)
\(\int \ln (\sqrt x)dx\)

Expert Answers (2)

2021-05-03
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17
 
Best answer
2021-09-09

Solution:

\(\int\ln(\sqrt{x})dx=\int1\cdot\ln(\sqrt{x})dx\)

Now we apply integration by parts

\(=[x\cdot\ln(\sqrt{x})]-\int\frac{1}{\sqrt{x}}(\frac{1}{2\sqrt{x}})\cdot xdx+C\)

\(=[x\ln(\sqrt{x})]-\int\frac{1}{2x}\cdot xdx+C\)

\(=x\ln\sqrt{x}-\frac{1}{2}\int dx+C\)

\(\int\ln(\sqrt{x})dx=x\ln\sqrt{x}-\frac{x}{2}+C\)

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