Maharashtra Board Class 12 Maths Part 2 Chapter 3 Indefinite Integration PDF Download

3. Indefinite Integration

Let us Study

Definition and Properties

Different Techniques: 1. by substitution 2. by parts 3. by partial fraction

Introduction

In differential calculus, we studied differentiation or derivatives of some functions. We saw that derivatives are used for finding the slopes of tangents, maximum or minimum values of the function.

Now we will try to find the function whose derivative is known, or given f(x). We will find g(x) such that g'(x) = f(x). Here the integration of f(x) with respect to x is g(x) or g(x) is called the primitive of f(x). For example, we know that the derivative of \(x^3\) with respect to x is \(3x^2\). So \(\frac{d}{dx} x^3 = 3x^2\); and integral of \(3x^2\) with respect to x is \(x^3\). This is shown with the sign of integration namely \(\int\). We write \(\int 3x^2 \cdot dx = x^3\).

In this chapter we restrict ourselves only to study the methods of integration. The theory of integration is developed by Sir Isaac Newton and Gottfried Leibnitz.

\(\int f(x) \cdot dx = g(x)\) read as an integral of f(x) with respect to x is g(x). Since the derivative of constant function with respect to x is zero (0), we can also write

\(\int f(x) \cdot dx = g(x) + c\) where c is an arbitrary constant and c can take infinitely many values.

For example

f(x) = \(x^2\) + c represents family of curves for different values of c

f'(x) = 2x gives the slope of the tangent to f(x) = \(x^2\) + c

In the figure we have shown the curves y = \(x^2\), y = \(x^2\) + 4, y = \(x^2\) - 5.

Note that at the points (2, 4), (2, 8), (2, -1) respectively on those curves, the slopes of tangents are 2(2) = 4.

Teacher's Note

Integration is the opposite of differentiation. Just like we find the slope of a curve using derivatives, we find the area under a curve using integration.

Exam Trick

Always remember to add the constant c at the end. The constant c is like the "extra knowledge" - it shows all possible curves that have the same slope at each point.

Points to Remember

Integration is the reverse of differentiation.
The symbol \(\int\) means "integral".
The constant c must always be added to indefinite integrals.
Different curves can have the same derivative at every point, differing only by a constant.

3.1.1 Elementary Integration Formulae

(i) \(\frac{d}{dx}\left(\frac{x^{n+1}}{n+1}\right) = (n+1)x^n\), \(n \neq -1\) \(\Rightarrow \int x^n \cdot dx = \frac{x^{n+1}}{n+1} + c\)

\(\frac{d}{dx}\left(\frac{(ax+b)^{n+1}}{(n+1) \cdot a}\right) = (ax+b)^n\), \(n \neq -1\) \(\Rightarrow \int (ax+b)^n \cdot dx = \frac{(ax+b)^{n+1}}{n+1} \cdot \frac{1}{a} + c\)

This result can be extended for n replaced by any rational \(\frac{p}{q}\).

(ii) \(\frac{d}{dx}\left(\frac{a^x}{\log a}\right) = a^x, a > 0\) \(\Rightarrow \int a^x \cdot dx = \frac{a^x}{\log a} + c\)

\(\int A^{ax+b} \cdot dx = \frac{A^{ax+b}}{\log A} \cdot \frac{1}{a} + c\), \(A > 0\)

(iii) \(\frac{d}{dx} e^x = e^x\) \(\Rightarrow \int e^x \cdot dx = e^x + c\)

\(\int e^{ax+b} \cdot dx = e^{ax+b} \cdot \frac{1}{a} + c\)

(iv) \(\frac{d}{dx} \sin x = \cos x\) \(\Rightarrow \int \cos x \cdot dx = \sin x + c\)

\(\int \cos(ax+b) \cdot dx = \sin(ax+b) \cdot \frac{1}{a} + c\)

(v) \(\frac{d}{dx} \cos x = -\sin x\) \(\Rightarrow \int \sin x \cdot dx = -\cos x + c\)

\(\int \sin(ax+b) \cdot dx = -\cos(ax+b) \cdot \frac{1}{a} + c\)

(vi) \(\frac{d}{dx} \tan x = \sec^2 x\) \(\Rightarrow \int \sec^2 x \cdot dx = \tan x + c\)

\(\int \sec^2(ax+b) \cdot dx = \tan(ax+b) \cdot \frac{1}{a} + c\)

(vii) \(\frac{d}{dx} \sec x = \sec x \cdot \tan x\) \(\Rightarrow \int \sec x \cdot \tan x \cdot dx = \sec x + c\)

\(\int \sec(ax+b) \cdot \tan(ax+b) \cdot dx = \sec(ax+b) \cdot \frac{1}{a} + c\)

(viii) \(\frac{d}{dx} \csc x = -\csc x \cdot \cot x\) \(\Rightarrow \int \csc x \cdot \cot x \cdot dx = -\csc x + c\)

\(\int \csc(ax+b) \cdot \cot(ax+b) \cdot dx = -\csc(ax+b) \cdot \frac{1}{a} + c\)

(ix) \(\frac{d}{dx} \cot x = -\csc^2 x\) \(\Rightarrow \int \csc^2 x \cdot dx = -\cot x + c\)

\(\int \csc^2(ax+b) \cdot dx = -\cot(ax+b) \cdot \frac{1}{a} + c\)

(x) \(\frac{d}{dx} \log x = \frac{1}{x}, x > 0\) \(\Rightarrow \int \frac{1}{x} dx = \log x + c, x \neq 0\)

Also \(\int \frac{1}{(ax+b)} \cdot dx = \log(ax+b) \cdot \frac{1}{a} + c\)

We assume that the trigonometric functions and logarithmic functions are defined on the respective domains.

Teacher's Note

These formulas are very useful. In India, students use these formulas every day in higher mathematics. Learn them by heart.

Exam Trick

Remember: When integrating \((ax+b)^n\), divide by a at the end. It is like the chain rule in reverse!

Points to Remember

Power rule: \(\int x^n dx = \frac{x^{n+1}}{n+1} + c\) (when n ≠ -1).
For \(\int \frac{1}{x} dx\), the answer is always log x.
Exponential: \(\int e^x dx = e^x + c\).
Sine and cosine swap: \(\int \sin x dx = -\cos x + c\).
Trigonometric integrals have different formulas for each function.

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