Indefinite Integrals

Introduction

     Integration and differentiation are a pair of inverse operations. So far, from a given function, we have been finding its derivative but the question arises: what is the function whose derivative is known? If the derivative of a function is given, then the function itself is called anti-derivative or integral.

Indefinite Integrals as Antiderivative

     The expression ∫ f(x) dx

     is read "the indefinite integral of f(x) with respect to x," and stands for the set of all antiderivatives of f.

Indefinite Integrals as Antiderivative (Contd...)

     Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent.

     Comparison between differentiation and integration:

     1. The derivative of a function, when it exists is a unique function. The integral of a function is not so. However, it always differs by a constant only.

Integration by Substitution

     If u is a function of x, we can use the following formula to evaluate an integral.
∫f dx = ∫ (f/(du/dx)) du
Using the Formula Use of the formula is equivalent to the following procedure:
1. Write u as a function of x.

Some Special Integrals

     Prove that:

Integration by Partial Fraction

     Rational function: If P(x) and Q(x) are two polynomials in x, then the ratio of two polynomials, P(x) / Q(x) is called a rational function, where Q(x) is not equal to zero.

     Proper rational function: If the degree of the numerator of the rational function is less than that of the denominator, the rational function is called a proper rational function.

Integration by Parts

     In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. The rule arises from the product rule of differentiation.

     The formula for Integration by Parts is:

Some Special Types of Integrals

     Prove that

     The quadratic expression ax² + bx + c can be expressed in the form a(x²  ± A²) by the method of completing the square. The integrals can be evaluated by using the special integrals.

Some Special Types of Integrals (Contd...)

     Evaluate the integral

Summary

     

     f(x) is called the integrand, F(x) is called the particular integral and C the constant of integration.

Conclusion

     In this chapter we have learnt the definition of antiderivative and various techniques of integration.