Integration and applications
Integration and applications tests indefinite and definite integration, the use of substitution and integration by parts, integration of rational functions via partial fractions, and the geometric applications: area under a curve, area between curves and volumes of revolution. The trapezium rule also appears as a numerical approximation.
Standard integrals and techniques
Standard results include ∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1, ∫(1/x) dx = ln|x| + C, ∫eˣ dx = eˣ + C, ∫sin x dx = −cos x + C and ∫cos x dx = sin x + C. For ∫f′(x)/f(x) dx the answer is ln|f(x)| + C. Substitution reverses the chain rule; integration by parts reverses the product rule with ∫u dv = uv − ∫v du.
Definite integrals and areas
A definite integral is evaluated by substituting the limits into the antiderivative. Area beneath a curve is the integral of the positive function; where the curve crosses the axis, split the interval and add the magnitudes. Area between two curves is the integral of (upper − lower).
Volumes and numerical approximation
Rotating y = f(x) about the x-axis through 2π generates volume π∫y² dx between the limits. The trapezium rule approximates an integral using h/2[y₀ + 2(y₁ + … + yₙ₋₁) + yₙ] with h the strip width; it overestimates for a curve that is concave up.
Worked example
Find ∫x e^(2x) dx.
Use parts with u = x and dv = e^(2x) dx, so du = dx and v = ½e^(2x). Then the integral is ½x e^(2x) − ∫½e^(2x) dx = ½x e^(2x) − ¼e^(2x) + C.
Traps that cost marks
- Omitting the constant of integration in an indefinite integral.
- Failing to convert the limits when using substitution on a definite integral.
- Treating signed area as geometric area when part of the curve lies below the axis.
- Choosing the wrong function as u in integration by parts; the standard order is logarithmic, algebraic, trigonometric, exponential.