Integrals — Study Notes

1. Introduction

Integration is the inverse operation of differentiation. If differentiation finds the rate of change of a function, integration reconstructs the function from its rate of change, and simultaneously computes the area enclosed by a curve. The Fundamental Theorem of Calculus unifies these two viewpoints: the definite integral is the net area under a curve, and it can be evaluated by antidifferentiation.

This chapter introduces integration as the inverse of differentiation, presents the standard formulae for indefinite integrals, and develops the three central techniques: integration by substitution, integration by partial fractions, and integration by parts. It then defines the definite integral as the limit of a sum, establishes its properties, and applies the fundamental theorem to evaluate definite integrals. The chapter closes with the reduction-based evaluation of some special definite integrals.

Integration is the most computation-heavy chapter of the syllabus and carries a very high weightage in the board examination. The ability to recognise the form of an integrand and select the right technique quickly is the difference between a good and an excellent performance. The techniques developed here also feed directly into the chapter on application of integrals, where definite integrals are used to compute areas and volumes.

2. Integration as the Inverse of Differentiation

If $\frac{d}{dx}F(x) = f(x)$, then $\int f(x)\,dx = F(x) + C$, where C is the constant of integration. The constant appears because differentiation destroys constants; it is the reason every indefinite integral carries a + C.

Standard Integrals

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1), \qquad \int \frac{1}{x} dx = \ln|x| + C$$

$$\int e^x dx = e^x + C, \qquad \int \sin x\, dx = -\cos x + C, \qquad \int \cos x\, dx = \sin x + C$$

$$\int \sec^2 x\, dx = \tan x + C, \qquad \int \sec x \tan x\, dx = \sec x + C$$

$$\int \csc^2 x\, dx = -\cot x + C, \qquad \int \csc x \cot x\, dx = -\csc x + C$$

$$\int \frac{1}{\sqrt{1 - x^2}} dx = \sin^{-1}x + C, \qquad \int \frac{1}{1 + x^2} dx = \tan^{-1}x + C$$

Rules of Integration

• $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$
• $\int k f(x) dx = k \int f(x) dx$
• For the generalised power form, $\int [f(x)]^n f'(x) dx = \frac{[f(x)]^{n+1}}{n+1} + C$.

3. Integration by Substitution

The substitution method reverses the chain rule. If an integrand contains a function and its derivative (up to a constant factor), substitute $t = f(x)$, so that $dt = f'(x)dx$:

$$\int f'(x) \cdot g(f(x))\, dx = \int g(t)\, dt$$

Standard Substitution Forms

$$\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C$$

$$\int [f(x)]^n f'(x) dx = \frac{[f(x)]^{n+1}}{n+1} + C$$

$$\int \frac{f'(x)}{\sqrt{1 - (f(x))^2}} dx = \sin^{-1}(f(x)) + C$$

Common substitutions for trigonometric and radical expressions: $x = a\sin\theta$ for $\sqrt{a^2 - x^2}$, $x = a\tan\theta$ for $\sqrt{a^2 + x^2}$, and $x = a\sec\theta$ for $\sqrt{x^2 - a^2}$.

4. Integration by Partial Fractions

A rational function $\frac{P(x)}{Q(x)}$, where the degree of P is less than the degree of Q, can be decomposed into partial fractions whose integrals are logarithms:

For example:

$$\int \frac{1}{(x-1)(x-2)} dx$$

Decompose: $\frac{1}{(x-1)(x-2)} = \frac{A}{x-1} + \frac{B}{x-2}$. Solving gives $1 = A(x-2) + B(x-1)$; substituting x = 1 gives A = -1, and x = 2 gives B = 1. Hence

$$\int \frac{1}{(x-1)(x-2)} dx = -\ln|x-1| + \ln|x-2| + C = \ln\left|\frac{x-2}{x-1}\right| + C$$

5. Integration by Parts

The product rule in reverse gives

$$\int u\, dv = uv - \int v\, du$$

The function u is chosen according to the ILATE rule, in order of priority: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. The more complex function should usually be kept as u.

Special Cases

For $\int e^x [f(x) + f'(x)] dx = e^x f(x) + C$, a very useful shortcut.

Examples

$$\int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C = e^x(x - 1) + C$$

$$\int x \ln x\, dx = \frac{x^2}{2}\ln x - \int \frac{x^2}{2} \cdot \frac{1}{x} dx = \frac{x^2}{2}\ln x - \frac{x^2}{4} + C$$

6. Some Standard Integrals

The following integrals occur repeatedly and should be memorised, where a > 0:

$$\int \frac{dx}{a^2 + x^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C, \qquad \int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C$$

$$\int \frac{dx}{a^2 - x^2} = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C, \qquad \int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\left(\frac{x}{a}\right) + C$$

$$\int \frac{dx}{\sqrt{x^2 + a^2}} = \ln|x + \sqrt{x^2 + a^2}| + C, \qquad \int \frac{dx}{\sqrt{x^2 - a^2}} = \ln|x + \sqrt{x^2 - a^2}| + C$$

7. Definite Integrals

If f is continuous on [a, b] and F is an antiderivative of f, then

$$\int_a^b f(x)\, dx = F(b) - F(a)$$

Properties of Definite Integrals

1. $\int_a^b f(x) dx = -\int_b^a f(x) dx$
2. $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$
3. $\int_a^b f(x) dx = \int_a^b f(t) dt$ (the variable of integration is a dummy)
4. $\int_0^a f(x) dx = \int_0^a f(a - x) dx$
5. $\int_{-a}^{a} f(x) dx = 2\int_0^a f(x) dx$ if f is even, and $0$ if f is odd.
6. $\int_a^b f(x) dx = \int_a^b f(a + b - x) dx$

The even/odd property is a powerful simplification tool: integrals of odd functions over symmetric intervals are automatically zero.

Quick Revision Tables

Table 1: Standard Integrals

Table 2: Integration Techniques

Table 3: Definite Integral Properties

Mind Map

Important Diagrams (SVG)

Diagram 1: Area Interpretation of the Definite Integral

Diagram 2: Choosing u by the ILATE Rule in Integration by Parts

Common Mistakes

1. Forgetting the constant of integration C in indefinite integrals; its omission loses marks in every problem.
2. Using the power rule $\frac{x^{n+1}}{n+1}$ when n = -1; the integral of 1/x is ln|x|, not x^0/0.
3. In substitution, forgetting to change the limits of a definite integral; after substitution the new limits must be in terms of the new variable.
4. Applying partial fractions to a rational function whose numerator degree is not less than the denominator degree without first performing polynomial division.
5. Choosing the wrong function as u in integration by parts, leading to a more complicated integral; ILATE should guide the choice.
6. Forgetting absolute values in logarithmic integrals, such as $\int \frac{dx}{x} = \ln|x| + C$.
7. Writing $\int \frac{1}{1 + x^2} dx$ as $\sin^{-1}x$; the correct anti-derivative is $\tan^{-1}x$, since d/dx(tan^(-1)x) = 1/(1+x^2).
8. In definite integrals, evaluating F(a) - F(b) instead of F(b) - F(a); the upper limit comes first.
9. Misapplying the even/odd property when the interval is not symmetric about zero.
10. In integration by parts of definite integrals, forgetting to apply the limits to the whole expression uv and the remaining integral.

Exam Tips

1. Always write + C for indefinite integrals and substitute limits for definite integrals; this single habit preserves marks.
2. Learn to recognise the pattern $f'(x)[f(x)]^n$ instantly and substitute t = f(x); this covers a large class of board questions.
3. For rational functions, check the degree of numerator and denominator first; if needed, divide before decomposing into partial fractions.
4. In integration by parts, apply ILATE and for the special form $\int e^x[f(x) + f'(x)]dx$, use the shortcut $e^x f(x) + C$.
5. Use the property $\int_0^a f(x)dx = \int_0^a f(a-x)dx$ to evaluate integrals of the form $\int \frac{1}{1 + \tan x}$ over symmetric ranges.
6. Use the even/odd property to eliminate half of the work on symmetric intervals; odd functions integrate to zero over [-a, a].
7. For definite integrals with products, consider differentiating part of the integrand and integrating the rest, matching the pattern of the fundamental theorem.
8. Practise combining substitution with the standard forms $\sin^{-1}(x/a)$ and $\tan^{-1}(x/a)$; these appear frequently in both short and long answer questions.

Conclusion

Integration transforms the abstract idea of the anti-derivative into one of the most powerful computational tools in mathematics. The indefinite integral reconstructs functions from their rates of change, the substitution method and partial fractions and integration by parts provide systematic techniques for almost every standard integrand, and the definite integral supplies the area under a curve with the fundamental theorem linking the two worlds. The properties of definite integrals, especially the even-odd and shift properties, offer elegant shortcuts that save substantial time in examinations. Because integration threads through the application of integrals, differential equations, and beyond into physics and engineering, this chapter is rightly regarded as the heart of the calculus course. Diligent practice of recognising forms and choosing techniques is the surest route to mastery and to strong examination performance.