Question

find f(x) if f'(x)=[x^2+sqrt(x)]/x and f(1)=3

Answer 1

\[\begin{array}{l} \text{For the integrand }\frac{x^2+\sqrt{x}}{x}\text{, substitute }u=\sqrt{x}\text{ and }du=\frac{1}{2 \sqrt{x}}d x: \\ \text{}=2\int\limits \left(u^3+1\right) \, du \\ \text{Integrate the \sum term by term:} \\ \text{}=2\int\limits u^3 \, du+2\int\limits 1 \, du \\ \text{The \int\limits of }u^3\text{ is }\frac{u^4}{4}: \\ \text{}=\frac{u^4}{2}+2\int\limits 1 \, du \\ \text{The \int\limits of }1\text{ is }u: \\ \text{}=\frac{u^4}{2}+2 u+\text{c}\} \\ \text{Substitute back for }u=\sqrt{x}: \\ \text{}=\frac{x^2}{2}+2 \sqrt{x}+\text{c} \\\end{array}\]

Answer 2

\[f(x)=\frac{x^2}{2}+2 \sqrt{x}+c\]
\[f(1)=3\]
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\[f(1)=\frac{1^2}{2}+2 \sqrt{1}+c=3\]
\[c=\frac{1}{2}\]
\[f(x)=\frac{x^2}{2}+2 \sqrt{x}+\frac{1}{2}\]