Integration
Integral Calculus
\[
\int f(x)\,dx = F(x) + c
\]
Integral calculus is mainly concerned with summing the values of a function over a particular range, and is particularly useful for finding the area of an irregular shape.
Integration is the process of finding the anti-derivative.
\( \int f(x)\,dx \) means the indefinite integral of the function is to be calculated with respect to x.
The anti-derivative F(x) is called the integral, c is called the constant of integration.
\( F'(x) = f(x) \) so differentiating the integral results in the original function.
Basic Rules of Integration
Power Rule
\[
\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \qquad (n \ne -1)
\]
Examples
\[
\int x^3\,dx = \frac{x^4}{4} + C
\]
\[
\int (5x^4 - 3x^{-2})\,dx
= x^5 + 3x^{-1} + C
\]
Sum Rule
\[
\int \bigl(f(x) + g(x)\bigr)\,dx
=
\int f(x)\,dx
+
\int g(x)\,dx
\]
Examples
\[
\int (4x - 2)\,dx = 4\int x\,dx - \int 2\,dx
\]
\[
= 4\int x\,dx - \int 2x^{0}\,dx
\]
\[
= 4\int x\,dx - 2\int x^{0}\,dx
\]
\[
= 4\left(\frac{x^{2}}{2}\right) - 2x
\]
\[
= 2x^{2} - 2x + c
\]
\[
\int (3x + 5)\,dx = 3\int x\,dx + \int 5\,dx
\]
\[
= 3\left(\frac{x^{2}}{2}\right) + 5x
\]
\[
= \frac{3}{2}x^{2} + 5x + c
\]
Constant Rule
\[
\int k\,dx = kx + C
\]
Examples
\[
\int 7\,dx = 7x + C
\]
\[
\int (7x^6 + 4x^3 - 9)\,dx
= x^7 + x^4 - 9x + C
\]
\[
\int \bigl(5t^{3} - 8t\bigr)\,dt
= 5\int t^{3}\,dt - 8\int t\,dt
\]
\[
= 5\left(\frac{t^{4}}{4}\right) - 8\left(\frac{t^{2}}{2}\right) + c
\]
\[
= \frac{5t^{4}}{4} - 4t^{2} + c
\]
Generalised Power Rule (Linear Inner Function)
\[
\int (ax + b)^{n}\,dx
=
\frac{(ax + b)^{\,n+1}}{a(n+1)} + c
\qquad (n \neq -1)
\]
Examples
\[
\int (3x - 4)^{5}\,dx
\]
\[
= \frac{(3x - 4)^{6}}{3 \cdot 6} + c
\]
\[
= \frac{(3x - 4)^{6}}{18} + c
\]
\[
\int \sqrt{4x + 4}\,dx
\]
\[
= \int (4x + 4)^{1/2}\,dx
\]
\[
= \frac{(4x + 4)^{3/2}}{4 \cdot \tfrac{3}{2}} + c
\]
\[
= \frac{(4x + 4)^{3/2}}{6} + c
\]
\[
= \frac{\sqrt{(4x + 4)^{3}}}{6} + c
\]
Integrating Trig Functions
Sine
\[
\int \sin x\,dx = -\cos x + C
\]
Example
\[
\int \sin(3x)\,dx = -\frac{1}{3}\cos(3x) + C
\]
Sine
\[
\int \sin(ax + b)\,dx
= -\frac{1}{a}\cos(ax + b) + c
\]
Example
\[
\int \sin(3x - 2)\,dx
= -\frac{1}{3}\cos(3x - 2) + c
\]
Cosine
\[
\int \cos x\,dx = \sin x + C
\]
Example
\[
\int \cos(5x)\,dx = \frac{1}{5}\sin(5x) + C
\]
Cosine
\[
\int \cos(ax + b)\,dx
= \frac{1}{a}\sin(ax + b) + c
\]
Example
\[
\int \cos(5x + 6)\,dx
= \frac{1}{5}\sin(5x + 6) + c
\]
Tangent
\[
\int \tan x\,dx
= -\ln|\cos x| + C
\]
\[
= \ln|\sec x| + C
\]
Examples
\[
\int 4\tan x\,dx
= 4\int \tan x\,dx
\]
\[
= -4\ln|\cos x| + C
\]
\[
\int \tan(3x - 1)\,dx
\]
\[
= -\frac{1}{3}\ln|\cos(3x - 1)| + C
\]
\[
\int \bigl(\tan x + \sec^2 x\bigr)\,dx
\]
\[
= -\ln|\cos x| + \tan x + C
\]
Sec² and Cosec²
\[
\int \sec^2 x\,dx = \tan x + C
\]
\[
\int \csc^2 x\,dx = -\cot x + C
\]
Examples
\[
\int 3\sec^{2} x\,dx
= 3\tan x + C
\]
\[
\int 7\sec^{2} x\,dx
= 7\tan x + C
\]
\[
\int 5\csc^{2} x\,dx
= -5\cot x + C
\]
\[
\int 4\csc^{2} x\,dx
= -4\cot x + C
\]
\[
\int \bigl(3\sec^{2} x - 2\csc^{2} x\bigr)\,dx
\]
\[
= 3\tan x + 2\cot x + C
\]
Exponentials & Logarithms
Exponential Functions
\[
\int e^x\,dx = e^x + C
\]
\[
\int a^x\,dx = \frac{a^x}{\ln a} + C
\]
Examples
\[
\int 3e^{2x}\,dx = \frac{3}{2}e^{2x} + C
\]
\[
\int 5e^{-3x}\,dx = -\frac{5}{3}e^{-3x} + C
\]
Logarithmic Forms
\[
\int \frac{1}{x}\,dx = \ln|x| + C
\]
Examples
\[
\int \frac{3x^2}{x^3+1}\,dx
= \ln|x^3+1| + C
\]
\[
\int x e^{x^2}\,dx
= \frac{1}{2}e^{x^2} + C
\]
\[
\int_{1}^{3} 2^{x}\,dx
\]
\[
= \left[\frac{2^{x}}{\ln 2}\right]_{1}^{3}
\]
\[
= \frac{2^{3}}{\ln 2} \;-\; \frac{2^{1}}{\ln 2}
\]
\[
= \frac{8 - 2}{\ln 2}
\]
\[
= \frac{6}{\ln 2}
\]
Integration by Substitution
Substitution Method — General Strategy
When faced with an integral of the form
\[
\int g\bigl(f(x)\bigr)\, f'(x)\,dx
\]
proceed as follows:
\[
\text{1) Identify the inner function } f(x)
\]
\[
\text{2) Let } u = f(x) \quad\Rightarrow\quad du = f'(x)\,dx
\]
\[
\text{3) Substitute into the integral to rewrite it in terms of } u
\]
\[
\text{4) Integrate with respect to } u
\]
\[
\text{5) Substitute back: replace } u \text{ with } f(x)
\]
Example
\[
\int 3x^{2}(x^{3} - 4)^{4}\,dx
\]
\[
= \int (x^{3} - 4)^{4} \cdot 3x^{2}\,dx
\]
\[
\text{Let } u = x^{3} - 4 \quad\Rightarrow\quad du = 3x^{2}\,dx
\]
\[
\text{Substitute for } u
\]
\[
\int (x^{3} - 4)^{4} \cdot 3x^{2}\,dx
= \int u^{4}\,du
\]
\[
= \frac{u^{5}}{5} + C
\]
\[
\text{Swap back for } x
\]
\[
= \frac{1}{5}(x^{3} - 4)^{5} + C
\]
General Form
\[
\int f(g(x))\,g'(x)\,dx = \int f(u)\,du
\]
Example
\[
\int (3x^2)(x^3+1)^5\,dx
\]
\[
u = x^3 + 1,\quad du = 3x^2\,dx
\]
\[
\int u^5\,du = \frac{u^6}{6} + C
\]
\[
= \frac{(x^3+1)^6}{6} + C
\]
Logarithmic Substitution
Example
\[
\int \frac{2x}{x^2+4}\,dx
\]
\[
u = x^2 + 4,\quad du = 2x\,dx
\]
\[
\int \frac{1}{u}\,du = \ln|u| + C
\]
\[
= \ln(x^2+4) + C
\]
Trig Substitution
Example
\[
\int \cos^2 x\,\sin x\,dx
\]
\[
u = \cos x,\quad du = -\sin x\,dx
\]
\[
-\int u^2\,du = -\frac{u^3}{3} + C
\]
\[
= -\frac{\cos^3 x}{3} + C
\]
Example
\[
\int_{0}^{1/2} \frac{x}{\sqrt{1 - x^{2}}}\,dx
\]
\[
\text{Let } u = 1 - x^{2}
\quad\Rightarrow\quad
du = -2x\,dx
\]
\[
x\,dx = -\frac{1}{2}\,du
\]
\[
x = \frac{1}{2} \Rightarrow u = 1 - \left(\frac{1}{2}\right)^{2} = \frac{3}{4}
\]
\[
x = 0 \Rightarrow u = 1
\]
\[
\int_{0}^{1/2} \frac{x}{\sqrt{1 - x^{2}}}\,dx
= -\frac{1}{2}\int_{1}^{3/4} \frac{1}{\sqrt{u}}\,du
\]
\[
= -\frac{1}{2}\left[ 2u^{1/2} \right]_{1}^{3/4}
\]
\[
= -\left[ u^{1/2} \right]_{1}^{3/4}
\]
\[
= -\sqrt{\frac{3}{4}} + 1
\]
\[
= 1 - \frac{\sqrt{3}}{2}
\]
Common Forms
Linear Denominator
\[
\int \frac{1}{ax+b}\,dx = \frac{1}{a}\ln|ax+b| + C
\]
Examples
\[
\int \frac{3}{2x+5}\,dx
= \frac{3}{2}\ln|2x+5| + C
\]
Arctan Form
\[
\int \frac{1}{x^2+a^2}\,dx
= \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C
\]
Examples
\[
\int \frac{5}{x^2+9}\,dx
= \frac{5}{3}\arctan\left(\frac{x}{3}\right) + C
\]
Arcsin Form
\[
\int \frac{1}{\sqrt{a^2 - x^2}}\,dx
= \arcsin\left(\frac{x}{a}\right) + C
\]
Example
\[
\int \frac{1}{\sqrt{25 - x^2}}\,dx
= \arcsin\left(\frac{x}{5}\right) + C
\]
Books
Printed resources available at Amazon
Basic Integration
Basic Integration (Calculus Revision)
A quick revision guide to basic integration, designed for the SQA Higher Mathematics course, with additional material extending into Advanced Higher.
- Integral calculus
- Basic rules of integration
- Fundamental theorem of calculus
- Trig functions
- Logarithms and exponentials
- Integration by substitution
- Common forms
- Areas on graphs
- Newton’s equations of motion
- Areas under curves
- Integration and the area function
- Area between curve and the y‑axis
- Areas enclosed by the graph and the x‑axis
- Area between two graphs
As an Amazon Associate I earn from qualifying purchases.
© Alexander Forrest