Integration
Integration
Integration is examined as reverse differentiation and as area. Expect standard integrals, definite integrals evaluated with correct limits, substitution, integration by parts, partial fractions, and the trapezium rule as a numerical alternative.
Core rules
- ∫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.
- ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C.
- Definite integral: antiderivative at the upper limit minus the lower limit; area below the x-axis comes out negative, so split the interval or take the modulus as instructed.
- Substitution: replace dx by du/(du/dx) and change the limits too when the integral is definite.
- By parts: ∫u dv = uv − ∫v du; choose u as the part that simplifies when differentiated.
- Trapezium rule: h/2[y₀ + 2(y₁ + … + y_(n−1)) + y_n], with h = (b − a)/n.
Worked example
Find ∫x e^(2x) dx. Take u = x and dv = e^(2x) dx, so du = dx and v = e^(2x)/2. Then the integral is x e^(2x)/2 − ∫(e^(2x)/2) dx = x e^(2x)/2 − e^(2x)/4 + C. Differentiating the answer back gives e^(2x)/2 + x e^(2x) − e^(2x)/2 = x e^(2x), confirming the result.
Traps
- Omitting + C for indefinite integrals.
- Forgetting to change limits after substitution.
- Dropping the factor 1/k when integrating e^(kx).
- Adding areas of regions below the axis without taking absolute values.