Integrals — chapter notes

If $\dfrac{d}{dx}F(x) = f(x)$ on an interval, $F$ is an antiderivative of $f$ and $\displaystyle\int f(x)\,dx = F(x) + C$, where $C$ is the constant of integration. Two antiderivatives of the same function differ by a constant, so the graphs of $F(x) + C$ form a family of vertically shifted curves.

Integration is linear: $\displaystyle\int [kf \pm g]\,dx = k\int f\,dx \pm \int g\,dx$. Every derivative rule gives an integration rule (see the table). A given point on the curve fixes $C$.

Example: $\displaystyle\int \left(6x^2 - \dfrac4x + 5\cos x\right)dx = 2x^3 - 4\log|x| + 5\sin x + C$.

If the integrand contains a function $g(x)$ together with its derivative $g'(x)$, put $t = g(x)$, $dt = g'(x)\,dx$, integrate in $t$ and substitute back.

Trigonometric integrals often need an identity first: $\sin^2 x = \dfrac{1 - \cos 2x}{2}$, $\cos^2 x = \dfrac{1 + \cos 2x}{2}$, $\tan^2 x = \sec^2 x - 1$; for odd powers such as $\sin^3 x$ keep one factor for $dt$.

Useful results: $\displaystyle\int \tan x\,dx = \log|\sec x| + C$, $\displaystyle\int \cot x\,dx = \log|\sin x| + C$, $\displaystyle\int \sec x\,dx = \log|\sec x + \tan x| + C$, $\displaystyle\int \operatorname{cosec} x\,dx = \log|\operatorname{cosec} x - \cot x| + C$.

For $a > 0$ learn the six results in the table. For $\displaystyle\int \dfrac{dx}{ax^2 + bx + c}$ or $\displaystyle\int \dfrac{dx}{\sqrt{ax^2 + bx + c}}$, complete the square and use the matching form.

For $\displaystyle\int \dfrac{px + q}{ax^2 + bx + c}\,dx$, write $px + q = A\dfrac{d}{dx}(ax^2 + bx + c) + B$: the $A$ part gives $A\log|ax^2 + bx + c|$, the $B$ part is a special form. With a square root in the denominator, the $A$ part gives $2A\sqrt{ax^2 + bx + c}$.

Example: $\displaystyle\int \dfrac{dx}{x^2 - 8x + 25} = \dfrac13\tan^{-1}\dfrac{x - 4}{3} + C$.

A rational function $\dfrac{P(x)}{Q(x)}$ is proper if $\deg P < \deg Q$; otherwise divide first. A proper function is split according to the factors of $Q(x)$ (table). Find the constants by putting convenient values of $x$ (zeros of the factors) and by comparing coefficients.

Each $\dfrac{A}{x - a}$ integrates to $A\log|x - a|$; each $\dfrac{B}{(x - a)^2}$ to $-\dfrac{B}{x - a}$; $\dfrac{Bx + C}{x^2 + bx + c}$ splits into a log part and an inverse tangent part.

Example: $\dfrac{x + 5}{(x - 1)(x + 2)} = \dfrac{2}{x - 1} - \dfrac{1}{x + 2}$.

$\displaystyle\int u\,dv = uv - \int v\,du$. Choose $u$ in the order inverse trigonometric, logarithmic, algebraic, trigonometric, exponential. Sometimes the formula is used twice, or the original integral reappears and is solved for (as in $\displaystyle\int e^{ax}\sin bx\,dx$).

Special pattern: $\displaystyle\int e^x[f(x) + f'(x)]\,dx = e^xf(x) + C$.

Results proved by parts (table): $\displaystyle\int \sqrt{a^2 - x^2}\,dx$, $\displaystyle\int \sqrt{x^2 + a^2}\,dx$, $\displaystyle\int \sqrt{x^2 - a^2}\,dx$.

Fundamental theorem of calculus (stated without proof): if $f$ is continuous on $[a, b]$ and $F' = f$, then $\displaystyle\int_a^b f(x)\,dx = F(b) - F(a) = \left[F(x)\right]_a^b$. No constant is needed. For $f \ge 0$ the value is the area under the curve between $x = a$ and $x = b$.

When substituting in a definite integral, change the limits too: $\displaystyle\int_0^1 2xe^{x^2}\,dx = \int_0^1 e^t\,dt = e - 1$.

Not in the 2026–27 syllabus: the definite integral as the limit of a sum.

• $\displaystyle\int_a^b f = -\int_b^a f$ and $\displaystyle\int_a^a f = 0$.
• $\displaystyle\int_a^b f = \int_a^c f + \int_c^b f$: split where the formula changes, e.g. $\displaystyle\int_0^4 |x - 1|\,dx = \dfrac12 + \dfrac92 = 5$.
• $\displaystyle\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$, and $\displaystyle\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx$: write $I$ both ways and add.
• $\displaystyle\int_{-a}^{a} f = 0$ if $f$ is odd; $\displaystyle\int_{-a}^{a} f = 2\int_0^a f$ if $f$ is even.
• $\displaystyle\int_0^{2a} f = 2\int_0^a f$ if $f(2a - x) = f(x)$, and $0$ if $f(2a - x) = -f(x)$.