Integral Calculus – JEE Mains Mathematics
1. Integral as an Anti-Derivative
- Integration is the reverse operation of differentiation, i.e., the anti-derivative of a function.
- If F'(x) = f(x), then ∫f(x) dx = F(x) + C, where C is the constant of integration.
- Integration is used to find the area under curves, and solve problems involving rates of change.
2. Fundamental Integrals
- Some basic integrals for algebraic, trigonometric, exponential, and logarithmic functions:
- ∫x^n dx = (x^(n+1))/(n+1) + C, for n ≠ -1
- ∫sin(x) dx = -cos(x) + C
- ∫cos(x) dx = sin(x) + C
- ∫e^x dx = e^x + C
- ∫1/x dx = ln|x| + C
3. Integration Techniques
3.1. Integration by Substitution
- If a function is complicated, substitution is used to simplify the integral.
- If u = g(x), then ∫f(g(x)) * g'(x) dx = ∫f(u) du
- Example: ∫x * cos(x²) dx → let u = x², then du = 2x dx.
3.2. Integration by Parts
- Integration by parts is based on the product rule of differentiation:
- ∫u dv = uv - ∫v du
- It is useful when the integral involves a product of two functions.
- Example: ∫x * e^x dx → u = x, dv = e^x dx, then du = dx, v = e^x.
3.3. Integration by Partial Fractions
- Used to integrate rational functions by expressing them as a sum of simpler fractions.
- Example: ∫(1/(x² - 1)) dx can be decomposed into partial fractions.
- The technique involves expressing the denominator as a product of linear factors and solving for constants.
3.4. Integration Using Trigonometric Identities
- Some integrals can be simplified using trigonometric identities, such as:
- sin²(x) = (1 - cos(2x))/2
- cos²(x) = (1 + cos(2x))/2
- Using these, integrals like ∫sin²(x) dx or ∫cos²(x) dx can be simplified and solved.
4. The Fundamental Theorem of Calculus
- It establishes the connection between differentiation and integration.
- The first part states that if f(x) is continuous on [a, b], then the function F(x) = ∫[a, x] f(t) dt is differentiable, and its derivative is f(x).
- The second part states that if F(x) is the antiderivative of f(x), then ∫[a, b] f(x) dx = F(b) - F(a).
5. Properties of Definite Integrals
- If f(x) is continuous on [a, b], then:
- ∫[a, b] f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x).
- ∫[a, b] f(x) dx = -∫[b, a] f(x) dx (reverse limits).
- ∫[a, b] [f(x) ± g(x)] dx = ∫[a, b] f(x) dx ± ∫[a, b] g(x) dx.
- ∫[a, b] c f(x) dx = c ∫[a, b] f(x) dx (constant factor rule).
6. Evaluation of Definite Integrals
- To evaluate a definite integral, find the antiderivative of the function and then substitute the limits of integration.
- Example: ∫[0, 1] (2x) dx = [x²] from 0 to 1 = 1² - 0² = 1.
7. Determining Areas of Regions Bounded by Curves
- Definite integrals can be used to find areas between curves.
- If a curve is defined by y = f(x) and bounded by x = a and x = b, the area between the curve and the x-axis is given by:
- Area = ∫[a, b] f(x) dx
- For regions bounded by two curves y = f(x) and y = g(x), the area between the curves is:
- Area = ∫[a, b] [f(x) - g(x)] dx