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  1. 4L'Hopital's rule, limit
  2. 6Limit of (1−cos⁡x)/x2(1 - \cos x)/x^2
  3. 16Implicit differentiation, folium
  4. 21Maximizing rectangle area
  5. 22Optimizing box volume
  6. 23Related rates, sphere volume
  7. 24Related rates, circle area
  8. 27Integral of ln⁡x\ln x by parts
  9. 28Integration by parts
  10. 30Integral of sin⁡2x\sin^2 x
  11. 35Area between x2x^2 and xx
  12. 37Volume by disk method
  13. 41Telescoping series sum
  14. 44Maclaurin series of cos⁡x\cos x
  15. 45Radius of convergence, ratio test
  16. 49Double integral, inner yy first
  17. 50Separable ODE dy/dx=2ydy/dx = 2y

27Integral of ln⁡x\ln x by parts

Evaluate ∫1eln⁡x dx\displaystyle \int_1^e \ln x\,dx.

Use integration by parts with u = ln x and dv = dx, giving ∫ln x dx = x ln x − x. Evaluating from 1 to e: (e·1 − e) − (0 − 1) = 0 + 1 = 1. So the integral equals 1.

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