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How to Integrate the Natural Logarithm

Recall the formula for integration by parts., Identify u and dv., Differentiate u and integrate dv., Substitute., Evaluate the second integral to reach the solution.

5 Steps 1 min read Medium

Step-by-Step Guide

  1. Step 1: Recall the formula for integration by parts.

    This formula is related to the product rule of derivatives. ∫udv=uv−∫vdu{\displaystyle \int u{\mathrm {d} }v=uv-\int v{\mathrm {d} }u}

  2. Step 2: Identify u and dv.

    Let u=ln⁡x{\displaystyle u=\ln x} and dv=dx.{\displaystyle {\mathrm {d} }v={\mathrm {d} }x.} , du=1xdx,v=x{\displaystyle {\mathrm {d} }u={\frac {1}{x}}{\mathrm {d} }x,v=x} , ∫ln⁡xdx=xln⁡x−∫x1xdx{\displaystyle \int \ln x{\mathrm {d} }x=x\ln x-\int x{\frac {1}{x}}{\mathrm {d} }x} , ∫ln⁡xdx=xln⁡x−x+C{\displaystyle \int \ln x{\mathrm {d} }x=x\ln x-x+C} Remember that integrating adds a constant C,{\displaystyle C,} since the derivative of a constant is zero.

  3. Step 3: Differentiate u and integrate dv.

  4. Step 4: Substitute.

  5. Step 5: Evaluate the second integral to reach the solution.

Detailed Guide

This formula is related to the product rule of derivatives. ∫udv=uv−∫vdu{\displaystyle \int u{\mathrm {d} }v=uv-\int v{\mathrm {d} }u}

Let u=ln⁡x{\displaystyle u=\ln x} and dv=dx.{\displaystyle {\mathrm {d} }v={\mathrm {d} }x.} , du=1xdx,v=x{\displaystyle {\mathrm {d} }u={\frac {1}{x}}{\mathrm {d} }x,v=x} , ∫ln⁡xdx=xln⁡x−∫x1xdx{\displaystyle \int \ln x{\mathrm {d} }x=x\ln x-\int x{\frac {1}{x}}{\mathrm {d} }x} , ∫ln⁡xdx=xln⁡x−x+C{\displaystyle \int \ln x{\mathrm {d} }x=x\ln x-x+C} Remember that integrating adds a constant C,{\displaystyle C,} since the derivative of a constant is zero.

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