Summary

(1) Definite integral as the limit of a sum.

1. $\displaystyle\int_a^bf(x)\,dx=\lim_{n\to\infty}\dfrac{b-a}{n}\sum_{r=1}^nf\!\left(a+\dfrac{(b-a)r}{n}\right)$
2. On $[0,1]$: $\displaystyle\int_0^1f(x)\,dx=\lim_{n\to\infty}\dfrac1n\sum_{r=1}^nf\!\left(\dfrac rn\right)$ (2) Properties of definite integrals.

(i) $\int_a^bf(x)\,dx=\int_a^bf(u)\,du$ (ii) $\int_b^af(x)\,dx=-\int_a^bf(x)\,dx$ (iii) $\int_a^bf(x)\,dx=\int_a^cf(x)\,dx+\int_c^bf(x)\,dx$ (iv) $\int_a^bf(x)\,dx=\int_a^bf(a+b-x)\,dx$ (v) $\int_0^af(x)\,dx=\int_0^af(a-x)\,dx$ (vi) $\int_0^{2a}f(x)\,dx=\int_0^a\big[f(x)+f(2a-x)\big]\,dx$ (vii) if $f(x)$ even, $\int_{-a}^af(x)\,dx=2\int_0^af(x)\,dx$ (ix) if $f(x)$ odd, $\int_{-a}^af(x)\,dx=0$ (x) if $f(2a-x)=f(x)$, $\int_0^{2a}f(x)\,dx=2\int_0^af(x)\,dx$ (xi) if $f(2a-x)=-f(x)$, $\int_0^{2a}f(x)\,dx=0$ (xii) if $f(a-x)=f(x)$, $\int_0^ax\,f(x)\,dx=\dfrac a2\int_0^af(x)\,dx$.

(3) Bernoulli's formula. $\displaystyle\int uv\,dx=uv_{(1)}-u^{(1)}v_{(2)}+u^{(2)}v_{(3)}-u^{(3)}v_{(4)}+\cdots$

(4) Reduction formulae.

(i) $\int_0^{\pi/2}\sin^nx\,dx=\int_0^{\pi/2}\cos^nx\,dx=\dfrac{(n-1)(n-3)\cdots2}{n(n-2)\cdots3}$ ($n$ odd) $=\dfrac{(n-1)(n-3)\cdots1}{n(n-2)\cdots2}\cdot\dfrac\pi2$ ($n$ even)

(ii) If $n,m$ both even: $\int_0^{\pi/2}\sin^mx\cos^nx\,dx=\dfrac{(m-1)(m-3)\cdots1\cdot(n-1)(n-3)\cdots1}{(m+n)(m+n-2)\cdots2}\cdot\dfrac\pi2$

(iii) If $n$ odd (any $m$): $\int_0^{\pi/2}\sin^mx\cos^nx\,dx=\dfrac{(n-1)(n-3)\cdots2}{(m+n)(m+n-2)\cdots(m+2)}$

(5) Gamma formulae.

(i) $\Gamma(n)=\displaystyle\int_0^\infty e^{-x}x^{n-1}\,dx=(n-1)!$ (ii) $\displaystyle\int_0^\infty e^{-ax}x^n\,dx=\dfrac{n!}{a^{n+1}}$

(6) Area of the region bounded by a curve and lines. …