Integration and Applications
The Integration and Applications domain tests the reverse of differentiation, the standard techniques that reduce unfamiliar integrands, and the geometric use of the definite integral. Questions normally pair a technique - substitution, integration by parts or partial fractions - with an area or a volume of revolution.
Core rules
- ∫x^n dx = x^(n+1)/(n + 1) + C for n ≠ -1; ∫(1/x) dx = ln|x| + C.
- ∫e^x dx = e^x + C; ∫sin x dx = -cos x + C; ∫cos x dx = sin x + C; ∫sec^2 x dx = tan x + C.
- By parts: ∫u dv = uv - ∫v du; choose u so that differentiating simplifies it - the usual order is logarithm, inverse trigonometric, algebraic, trigonometric, exponential.
- Substitution: change the variable and the limits together in a definite integral; never combine old limits with a new variable.
- Area between a curve and the x-axis from a to b is ∫|f(x)| dx; between two curves it is ∫(upper - lower) dx.
- Volume from rotation about the x-axis is π∫y^2 dx, with the limit handling exactly as for area.
Worked example
Evaluate ∫ from 0 to 1 of x e^x dx. Take u = x and dv = e^x dx, so du = dx and v = e^x. Then ∫x e^x dx = x e^x - ∫e^x dx = x e^x - e^x. Evaluating between 0 and 1 gives (1·e - e) - (0 - 1) = 0 + 1 = 1.
Traps
- Omitting + C in an indefinite integral, which loses a mark even when the antiderivative is correct.
- Writing ∫(1/x) dx = ln x, which fails for negative x; the modulus is required.
- Dropping the π and the square in a volume of revolution.
- Integrating straight across a root so that a region below the x-axis cancels a region above it.
- Leaving an answer in terms of the substituted variable when the question is posed in x.