Chapter 8 – Integral Calculus (JEE)
1. Introduction to Integration
Integration is the reverse process of differentiation.
If $\dfrac{d}{dx}[F(x)] = f(x)$, then:
$$\int f(x)\,dx = F(x) + C$$
$C$ is called the constant of integration.
2. Standard Integrals
| Function | Integral |
|---|---|
| $x^n$ | $\dfrac{x^{n+1}}{n+1}+C,\; n\neq -1$ |
| $\dfrac{1}{x}$ | $\ln|x|+C$ |
| $e^x$ | $e^x+C$ |
| $\sin x$ | $-\cos x+C$ |
| $\cos x$ | $\sin x+C$ |
3. Properties of Indefinite Integrals
$$\int [f(x)+g(x)]dx = \int f(x)dx + \int g(x)dx$$
$$\int kf(x)dx = k\int f(x)dx$$
4. Integration by Substitution
Used when integrand contains a function and its derivative.
If $t = g(x)$, then
$$\int f(g(x))g'(x)\,dx = \int f(t)\,dt$$
Always change limits also in definite integrals.
5. Integration by Parts
$$\int u\,dv = uv - \int v\,du$$
Choose $u$ using ILATE rule:
Inverse → Log → Algebra → Trigonometric → Exponential
6. Integration of Trigonometric Functions
Special identities are used:
- $\sin^2x = \dfrac{1-\cos2x}{2}$
- $\cos^2x = \dfrac{1+\cos2x}{2}$
7. Integration Using Partial Fractions
Used when integrand is a rational function:
$$\int \frac{P(x)}{Q(x)}dx$$
Applicable when degree of $P(x) <$ degree of $Q(x)$.
8. Definite Integral
A definite integral represents the net area under the curve.
$$\int_a^b f(x)\,dx = F(b) - F(a)$$
9. Properties of Definite Integrals
$$\int_a^a f(x)\,dx = 0$$
$$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$$
$$\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$$
10. Symmetry Properties
If $f(x)$ is:
- Even → $\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx$
- Odd → $\int_{-a}^{a} f(x)\,dx = 0$
11. Definite Integrals Using Substitution
$$\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$$
Very important JEE property.
12. Integration as Area Under Curve
Area between curve $y=f(x)$ and x-axis from $a$ to $b$ is:
$$\text{Area} = \int_a^b |f(x)|\,dx$$
13. Area Between Two Curves
$$\int_a^b [f(x)-g(x)]\,dx$$
Upper function − Lower function.
14. Common JEE Traps
- Forgetting constant of integration
- Wrong substitution limits
- Ignoring symmetry
- Incorrect choice of $u$ in parts
15. Typical JEE Question Patterns
| Pattern | Method |
|---|---|
| Polynomial integrals | Standard formula |
| Product of functions | Integration by parts |
| Rational functions | Partial fractions |
| Symmetric limits | Properties of definite integrals |
16. Final Revision Checklist
You are fully prepared if you can:
- Evaluate all standard integrals
- Apply substitution and parts confidently
- Use definite integral properties smartly
- Calculate areas correctly
- Avoid common mistakes