Definite Integrals — chapter notes

If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then $\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)$. The theorem is used without proof (its proof is not in the ISC 2027 syllabus).

• The constant of integration cancels, so it is never written.
• The integral counts area below the axis as negative.
• The theorem needs $f$ continuous on the closed interval: $\displaystyle\int_{-1}^{1} \dfrac{dx}{x^2}$ cannot be found this way.

If $t = g(x)$, then $\displaystyle\int_a^b f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(t)\,dt$.

Find the new limits by putting the old limits into $t = g(x)$, then finish in $t$. Do not return to $x$, and do not keep the old limits.

• P0 $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(t)\,dt$ (dummy variable)
• P1 $\displaystyle\int_a^b f = -\int_b^a f$; in particular $\displaystyle\int_a^a f = 0$
• P2 $\displaystyle\int_a^b f = \int_a^c f + \int_c^b f$

Modulus: split where the inside changes sign and remove the modulus on each piece with the correct sign.

Greatest integer $[x]$: split at every integer (or every point where the inside is an integer); on each piece the function is constant.

• P3 $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$
• P4 $\displaystyle\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx$

Proof (P4): put $t = a - x$; then $dx = -dt$, $x = 0 \Rightarrow t = a$, $x = a \Rightarrow t = 0$, so the integral becomes $-\displaystyle\int_a^0 f(t)\,dt = \int_0^a f(t)\,dt$.

Reflect and add: call the integral $I$, write the reflected form, add the two, and divide by $2$. On $\left[0, \tfrac{\pi}{2}\right]$, P4 swaps $\sin x$ and $\cos x$; it also shows $\displaystyle\int_0^{\pi/2} \log\tan x\,dx = 0$.

The factor $x$: $\displaystyle\int_0^{\pi} x f(\sin x)\,dx = \dfrac{\pi}{2}\int_0^{\pi} f(\sin x)\,dx$.

• P5 $\displaystyle\int_0^{2a} f(x)\,dx = \int_0^a f(x)\,dx + \int_0^a f(2a - x)\,dx$

So $\displaystyle\int_0^{2a} f = 2\int_0^a f$ if $f(2a - x) = f(x)$, and $\displaystyle\int_0^{2a} f = 0$ if $f(2a - x) = -f(x)$.

Examples: $\displaystyle\int_0^{\pi} \sin^5 x\,dx = 2\int_0^{\pi/2} \sin^5 x\,dx$; $\displaystyle\int_0^{\pi} \cos^3 x\,dx = 0$; $\displaystyle\int_0^{2\pi} \sin^3 x\,dx = 0$.

• Even: $f(-x) = f(x)$, e.g. $x^2$, $\cos x$, $|x|$; then $\displaystyle\int_{-a}^{a} f = 2\int_0^a f$.
• Odd: $f(-x) = -f(x)$, e.g. $x^3$, $\sin x$, $x\cos x$, $\log\dfrac{2 - x}{2 + x}$; then $\displaystyle\int_{-a}^{a} f = 0$.

Any integrand can be split into an even part and an odd part; integrate only the even part. The interval must be balanced: on $[-1, 2]$ these rules do not apply.