Indefinite integrals

If \[ \frac{d}{dx} \log x = \frac{1}{x} \quad \quad (\text{A}) \] find \[ \int \frac{1}{x} dx \]

Undo the differentiation that produced the rhs

\[ \begin{align} \int \frac{d}{dx} \log x dx &= \int \frac{1}{x} dx \\ \int \frac{1}{x} dx &= \log x \end{align} \ \]

Add the constant

\[ \begin{align} \int \frac{1}{x} x dx &= \log x + c \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

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Now You Try:

If \[ \frac{d}{dx} \sqrt x = \frac{1}{2 \sqrt x} \quad \quad (\text{A}) \] find \[ \int \frac{1}{2 \sqrt x} dx \]