Summary of Class 12 Maths Integrals

Summary of Class 12 Maths Integrals

Integration

Integration is the inverse process of differentiation. 

If y = F(x) has derivative dy/dx = f(x), then the integral of f(x) with respect to x, written as  is the function F(x). F(x) is called an anti-derivative of f(x). 

Some Basic Formulae

• ∫ 1 dx = x + C
• ∫ a dx = ax + C
• ∫ xⁿ dx = (xⁿ⁺¹)/(n + 1) + C ; n ≠ -1
• ∫ sin x dx = – cos x + C
• ∫ cos x dx = sin x + C
• ∫ tan x dx = log |sec x| + C
• ∫ cot x dx = log |sin x| + C
• ∫ sec x dx = log |sec x + tan x| + C
• ∫ cosec x dx = log |cosec x – cot x| + C
• ∫ sec² x dx = tan x + C
• ∫ cosec² x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec x cot x dx = – cosec x + C
• ∫ (1/x) dx = log |x| + C
• ∫ e^x dx = e^x + C
• ∫ a^x dx = (a^x/log a) + C; a > 0, a ≠ 1

Integrals of Some Special Functions

 

Methods of Integration

1. Integration by substitution

 Sometimes, it is easy to integrate f(x) by substituting t = g(x).

We should remember that dt/dx = g’(x) and so, dt = g’(x) dx. 

For example, consider I = ∫e ^x sin(e^x) dx.

Putting t = e^x, we obtain dt = e^x dx and so,

I = ∫e ^x sin(e^x) dx = ∫ sin t dt

                             = -cos t + C = -cos(e^x) + C

 (Remember to express your answer as a function of x.)

 

2. Integration using Partial Fractions

If the given integral is of the form ∫ p(x)/q(x) dx, where p(x) and q(x) are polynomials in x and q(x) ≠ 0, then we solve such integrals by partial fractions. We first express it as the sum of partial fractions and then integrate each term by using a suitable formula.

 

3. Integration by Parts

This is based on the product rule of differentiation.

Let u and v be two differentiable functions of a single variable x, then

∫ f(x) g(x) dx = f(x) ∫ g(x) dx – ∫ [f’(x) ∫ g(x) dx] dx

 

Definite Integral

Let F(x) be the integration of a continuous function f(x) defined on [a, b] , then the definite integral of f(x) over [a, b] is denoted by

  [F(b) – F(a)].

i.e.

The numbers a and b are called the limits of integration. ‘a’ is called the lower limit and ‘b’ is called the upper limit.

 

Evaluation of Definite Integrals

To evaluate the definite integral  of a continuous function f(x) defined on [a, b], we may use the following steps.

Steps:

Step I: Find the indefinite integral  Let this be F(x).

Step II: Evaluate F(b) and F(a).

Step III: Calculate F(b) – F(a).

The number obtained in step III is the value of the definite integral

 

Properties of Definite Integrals