Evaluate the integral. int(3-x)/(sqrt x)dx

Evaluate the integral. int(3-x)/(sqrt x)dx

Question
Applications of integrals
asked 2021-03-07
Evaluate the integral.
\(\displaystyle\int\frac{{{3}-{x}}}{{\sqrt{{x}}}}{\left.{d}{x}\right.}\)

Answers (1)

2021-03-08
\(\displaystyle\int\frac{{{3}-{x}}}{{\sqrt{{x}}}}{\left.{d}{x}\right.}\)
\(\displaystyle={3}\int\frac{{1}}{{\sqrt{{x}}}}{\left.{d}{x}\right.}-\int{\left(\sqrt{{x}}\right)}{\left.{d}{x}\right.}\)
\(\displaystyle={3}{\left[\frac{{{x}^{{-\frac{{1}}{{2}}}}+{1}}}{{-\frac{{1}}{{2}}+{1}}}\right]}-{\left[\frac{{{x}^{{\frac{{1}}{{2}}}}+{1}}}{{\frac{{1}}{{2}}+{1}}}\right]}+{c}\)
\(\displaystyle={3}{\left({2}\sqrt{{x}}\right)}-\frac{{2}}{{3}}{x}^{{\frac{{3}}{{2}}}}+{c}\)
\(\displaystyle\therefore\int\frac{{{3}-{x}}}{{\sqrt{{x}}}}{\left.{d}{x}\right.}={6}\sqrt{{x}}-\frac{{2}}{{3}}{x}^{{\frac{{3}}{{2}}}}+{c}\)
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