Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — integration by parts
Find ∫xexdx.
- 1∫udxdvdx=uv−∫vdxdudxThe reverse of the product rule.
- 2Choose u=x, since it simplifies to 1 when differentiated.The exponential is equally easy either way, so let the power be u.
- 3dxdv=ex, so v=ex.
- 4∫xexdx=xex−∫ex⋅1dxThe new integral is simpler than the original — the sign the choice was right.
- 5=xex−ex+C=ex(x−1)+CCheck by differentiating: ex(x−1)+ex=xex ✓
Example 2 — partial fractions
Find ∫(x−1)(x+2)5x+1dx.
- 1Split into partial fractions: (x−1)(x+2)5x+1≡x−1A+x+2BTwo distinct linear factors, so two constants.
- 2Multiply up: 5x+1≡A(x+2)+B(x−1)
- 3Let x=1: 6=3A, so A=2.Choosing the value that kills a bracket is far quicker than comparing coefficients.
- 4Let x=−2: −9=−3B, so B=3.
- 5∫(x−12+x+23)dxEach piece is now a standard log integral.
- 6=2ln∣x−1∣+3ln∣x+2∣+CModuli required — the arguments can be negative.
Example 3 — the trapezium rule
Estimate ∫01x2dx using the trapezium rule with 4 strips, and comment on the accuracy.
- 1h=nb−a=41−0=0.254 strips means 5 ordinates.
- 2Ordinates at x=0,0.25,0.5,0.75,1: 0, 0.0625, 0.25, 0.5625, 1
- 32h[y0+2(y1+y2+y3)+y4]Ends once, middles doubled.
- 4=0.125[0+2(0.875)+1]=0.125(2.75)
- 5=0.34375The exact value is 31=0.3333.
- 6It is an overestimate, because the curve bends upwards and the chords lie above it.This reasoning is worth a mark on its own.