Calculus: Integration and Applications
Calculus: Integration and Applications tests antiderivatives, definite integrals, areas and volumes. You must choose an effective method and track limits and signs.
Core rules
- Standard forms: ∫x^n dx=x^{n+1}/(n+1)+C for n≠-1; ∫1/x dx=ln|x|+C; ∫e^{kx} dx=e^{kx}/k+C.
- Substitution: change dx to (dx/du)du and change limits for definite integrals.
- Integration by parts: ∫u dv=uv-∫v du; choose u by a priority such as logarithm, inverse trig, algebraic, trig, exponential.
- Area between curves: ∫(upper-lower)dx. Volume of revolution about the x-axis: π∫y²dx.
Worked example
∫x ln x dx. Let u=ln x and dv=x dx. Then du=dx/x and v=x²/2. So the integral equals (x²/2)ln x - ∫(x²/2)(1/x)dx = (x²/2)ln x - x²/4 + C.
Traps
Forgetting du in substitution; not changing limits; dropping absolute value in ln|x|; using y² when rotating about the y-axis; adding C to a definite integral; ignoring area below the x-axis and losing a sign.