Question

integral of log x value is

Answer 1

Answer on Question# 54548– Mathematics – Integral Calculus

Question:

Integral of $\log x$ value is ...

Answer:

Let's use the following notation: $\log(x) \equiv \ln(x)$. Hence, we have

$$\begin{aligned} \int \ln x \, dx &= \left\{ \text{the integration by parts formula:} \int U(x) V'(x) \, dx = U(x) V(x) - \int V(x) U'(x) \, dx \right\} \\ &= \left\{ U(x) = \ln(x), \, U'(x) = \frac{1}{x}; \, V'(x) = 1, \, V(x) = x \right\} = x \ln(x) - \int x \cdot \frac{dx}{x} \\ &= x \ln(x) - \int dx = x \ln(x) - x + \text{Const} = x (\ln(x) - 1) + \text{Const}. \end{aligned}$$

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