Integrate the following by parts integral (lnx) dx

To solve the integral of the natural logarithm function, specifically the integral of ln(x) with respect to x, we will employ the method of integration by parts. This technique is based on the formula for integration by parts, which states that the integral of u dv can be expressed as u v minus the integral of v du. In this case, we will choose u to be ln(x) and dv to be dx. Consequently, we need to differentiate u to find du and integrate dv to find v.

Differentiating u, we have du = (1/x) dx, while integrating dv gives us v = x. Substituting these values into the integration by parts formula, we can express the integral of ln(x) dx as follows: ∫ln(x) dx = x ln(x) – ∫x (1/x) dx. The integral on the right simplifies to ∫1 dx, which is simply x. Therefore, we can rewrite the original integral as x ln(x) – x.

Finally, we must include the constant of integration to account for the indefinite nature of the integral. Thus, the complete solution to the integral of ln(x) dx is x ln(x) – x + C, where C represents the constant of integration. This result encapsulates the process of integration by parts applied to the natural logarithm function, providing a clear and concise expression for the integral.

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