silemus.
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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

30Integral of sin⁡2x\sin^2 x

Evaluate ∫0π/2sin⁡2x dx\displaystyle \int_0^{\pi/2} \sin^2 x\,dx. Give the answer to 3 decimal places.

Rewrite sin²x using the double angle identity (1−cos2x)/2, then integrate term by term. This gives π/4, which is about 0.785. Checked by noting sin²x averages to 1/2 over the interval, times the interval length π/2.

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