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Introduction |
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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. |
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Indefinite Integrals as Antiderivative |
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The expression ∫ f(x) dx |
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is read "the indefinite integral of f(x) with respect to x," and stands for the set of all antiderivatives of f. |
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Thus, ∫ f(x) dx is a collection of functions; it is not a single function, nor a number. The function f that is being integrated is called the integrand, and the variable x is called the variable of integration. (The expression dx is short for "with respect to x."). |
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Indefinite Integrals as Antiderivative (Contd...) |
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Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. |
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Comparison between differentiation and integration: |
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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. |
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2. The derivative of a function has a geometrical meaning, namely, the slope of the tangent to the corresponding curve at the point. Similarly, indefinite integral of a function represents geometrically, a family of curves placed parallel to each other having parallel tangents at the points of intersection of the curves of the family with the lines orthogonal to the axis representing the variable of integration. |
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3. Differentiation is a process involving limits, so is integration. |
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Integration by Substitution |
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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.
2. Take the derivative du/dx, and solve for the quantity dx in terms of du.
3. Use the expression you obtain in part 2 to substitute for dx in the given integral. |
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When the integrand consists of trigonometric function, we use suitable trigonometric identities to simplify the function so that it can be integrated. |
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Some Special Integrals |
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Prove that: |
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Integration by Partial Fraction |
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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. |
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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. |
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Improper rational function: If the degree of the numerator is greater than the degree of the denominator in a rational fraction, then the rational function is called improper rational function. |
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Partial fractions: Any proper rational function P(x)/Q(x) can be expressed as sum of rational fractions, each having a factor of Q(x). Each such fraction is known as Partial fraction. |
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Integration by Parts |
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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. |
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The formula for Integration by Parts is:
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Some Special Types of Integrals |
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Prove that  |
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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. |
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Some Special Types of Integrals (Contd...) |
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Evaluate the integral 
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Summary |
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f(x) is called the integrand, F(x) is called the particular integral and C the constant of integration. |
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 Method of substitution: |
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If the integrand f(x) of the integral is not in an integral form the variable of integration x is changed to a suitable variable z by substitution and on differentiation and simplification, the new integral is found integrable. |
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Conclusion |
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In this chapter we have learnt the definition of antiderivative and various techniques of integration.
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Welcome! 
