Maharashtra Board Class 12 Maths Commerce Part I Chapter 5 Integration PDF Download

Integration

Let's Study

Method of Substitution

Some Special Integrals

Integration by Parts

Integration by Partial Fraction

Let's Recall

Derivatives

5.1.1 Introduction

In this chapter, we shall study the operation which is an inverse process of differentiation. We now want to study the problem: the derivative of a function is given and we have to determine the function. The process of determining such a function is called integration.

Consider the following examples:

(1) Suppose we want to determine a function whose derivative is \(3x^2\). Since we know that \(\frac{d}{dx}(x^3) = 3x^2\), therefore the required function is \(f(x) = x^3\).

\(x^3\) is called integral of \(3x^2\) with respect to \(x\), and this is written as \(\int 3x^2 \, dx = x^3\).

The symbol \(\int\), called the integration sign, was introduced by Leibnitz. \(dx\) indicates that the integration is to be taken with respect to the variable \(x\).

(2) Suppose we want to determine a function whose derivative is \(\frac{1}{x}\). Since we know that \(\frac{d}{dx}(\log x) = \frac{1}{x}\), therefore the required function is \(\log x\). Using the integral sign, we can write \(\int \frac{1}{x} \, dx = \log x\), where \(x > 0\).

Let's Learn

Definition: Integral or primitive or antiderivative of a function.

If \(f(x)\) and \(g(x)\) are two functions such that \(\frac{d}{dx}[f(x)] = g(x)\), then \(f(x)\) is called an integral of \(g(x)\) with respect to \(x\). It is denoted by \(\int g(x) \, dx = f(x)\) and read as integral of \(g(x)\) with respect to \(x\) is \(f(x)\). Here, we say that \(g(x)\) is the integrand.

This process of finding the integral of a function is called integration. Thus, integration is the inverse operation of differentiation.

For example:

\(\frac{d}{dx}(x^4) = 4x^3\)

\(\therefore \int 4x^3 \, dx = x^4\)

But, note that:

\(\frac{d}{dx}(x^4 + 5) = 4x^3\)

\(\frac{d}{dx}(x^4 - 8) = 4x^3\)

What is the observation? Can you generalize from the observation?

In general:

\(\frac{d}{dx}(x^4 + c) = 4x^3\), where \(c\) is any real number.

Hence, in general, we write:

\(\therefore \int 4x^3 \, dx = x^4 + c\)

The number \(c\) is called constant of integration.

Teacher's Note

Integration is like finding what was before differentiation. It is the opposite of taking away. Just like if you break a chocolate bar into pieces, integration puts it back together.

Exam Trick

Always remember to add \(+ c\) at the end of every integration. This \(c\) means we do not know the exact number. Like having money but not knowing how much was in your pocket before counting.

Points to Remember

Integration is the reverse of differentiation.

The symbol \(\int\) means find the integral.

Always add a constant \(c\) to the final answer.

The thing under the integral sign is called the integrand.

The \(dx\) tells us which variable we are using.

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