Definite Integration — NDA Mathematics

Formulas (3)

• The Fundamental Theorem of Calculus · Fundamental Theorem of Calculus and its corollaries
  $$\int_a^b f(x)\,dx = F(b)-F(a)\;\;(F'=f) \qquad \int_a^b f'(x)\,dx = f(b)-f(a) \qquad \int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)|$$
• Integrals of periodic functions · Integral of a periodic function
  $$\int_0^{nT} f(x)\,dx = n\int_0^{T} f(x)\,dx \qquad \int_{a}^{a+nT} f(x)\,dx = n\int_0^{T} f(x)\,dx$$
• The Leibniz rule — differentiating an integral · Leibniz rule (variable upper limit)
  $$\frac{d}{dx}\int_{a}^{g(x)} f(t)\,dt = f(g(x))\,g'(x)$$

Watch out for (3)

• Integrating a derivative is not always trivial
  → The Fundamental Theorem of Calculus
• Use the function's period, not the trig argument's
  → Integrals of periodic functions
• Don't forget the chain-rule factor
  → The Leibniz rule — differentiating an integral

Formulas (3)

• King's property — the reflection trick · King's property and its companions
  $$\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx \qquad \int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx \qquad \int_0^{2a} f(x)\,dx = \int_0^a \big[f(x)+f(2a-x)\big]\,dx \qquad \int_0^a \frac{f(x)}{f(x)+f(a-x)}\,dx = \frac{a}{2}$$
• Odd/even symmetry over a symmetric interval · Odd/even symmetry over a symmetric interval
  $$\int_{-a}^{a} f(x)\,dx = 0 \;\;(f\text{ odd}) \qquad \int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx \;\;(f\text{ even}) \qquad \int_{-a}^{a} \frac{f(x)}{1+c^{x}}\,dx = \int_0^a f(x)\,dx \;\;(f\text{ even})$$
• Standard results and trig reductions · Standard definite-integral results
  $$\int_0^{\pi}\frac{dx}{1+\sin^2x} = \frac{\pi}{\sqrt2} \qquad \int_0^1 x^{m}(1-x)^{n}\,dx = \frac{m!\,n!}{(m+n+1)!} \qquad \int_0^{\pi/2}\sin^{2}x\,dx = \int_0^{\pi/2}\cos^{2}x\,dx = \frac{\pi}{4}$$

Watch out for (5)

• Add the reflected form — don't just substitute and stop
  → King's property — the reflection trick
• King's reflection only works with the matching limits
  → King's property — the reflection trick
• Check parity of the WHOLE integrand
  → Odd/even symmetry over a symmetric interval
• Reduce the power BEFORE integrating
  → Standard results and trig reductions
• Transform the limits when you substitute
  → Direct evaluation — simplify, then integrate

Formulas (1)

• Integrating greatest-integer (floor) functions · Greatest-integer (floor) results
  $$\int_n^{n+1}\big(x-\lfloor x\rfloor\big)\,dx = \frac12 \qquad \lfloor x\rfloor + \lfloor -x\rfloor = -1 \;\;(x\notin\mathbb{Z}) \qquad \int_0^{n}\lfloor x\rfloor\,dx = \frac{n(n-1)}{2}$$

Watch out for (3)

• Find the zeros first — don't drop the absolute value
  → Integrating absolute-value and piecewise functions
• Split |x| at 0, even when 0 is interior to the interval
  → Integrating absolute-value and piecewise functions
• ⌊x⌋ on [−1, 0) is −1, not 0
  → Integrating greatest-integer (floor) functions

Formulas (1)

• Area between curves · Area as a definite integral
  $$A = \int_a^b |f(x)|\,dx \qquad A = \int_a^b |f(x)-g(x)|\,dx$$

Watch out for (1)

• Area is unsigned — use the absolute value
  → Area between curves

Watch out for (1)

• One condition, one equation — match the counts
  → Solving for coefficients from integral conditions