53  Integration

53.1 Introduction

Integration is the inverse process of differentiation.
While differentiation deals with the rate of change, integration deals with the summation or accumulation of quantities.

It is widely used in mathematics, physics, engineering, and economics for:
- Finding areas under curves.
- Computing displacement from velocity.
- Solving differential equations.
- Evaluating growth and cumulative quantities.

Integration is of two main types:
1. Indefinite Integration → general antiderivative (includes constant of integration \(C\)).
2. Definite Integration → numerical value representing area or accumulation between limits.

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53.2 Indefinite Integration

53.2.1 Definition

If \(\frac{dy}{dx}=f(x)\), then
\[ y=\int f(x)\,dx + C \]
where \(C\) is the constant of integration.

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53.2.2 Standard Integrals

1. \(\int x^n dx=\frac{x^{n+1}}{n+1}+C \quad (n\neq -1)\)
   
2. \(\int \frac{1}{x}dx=\ln|x|+C\)
   
3. \(\int e^x dx=e^x+C\)
   
4. \(\int a^x dx=\frac{a^x}{\ln a}+C, \; a>0,a\neq 1\)
   
5. \(\int \sin x dx=-\cos x+C\)
   
6. \(\int \cos x dx=\sin x+C\)
   
7. \(\int \sec^2x dx=\tan x+C\)
   
8. \(\int \csc^2x dx=-\cot x+C\)
   
9. \(\int \sec x \tan x dx=\sec x+C\)
   
10. \(\int \csc x \cot x dx=-\csc x+C\)

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53.2.3 Rules of Integration

1. Linearity:
   \(\int [af(x)+bg(x)]dx=a\int f(x)dx+b\int g(x)dx\)

2. Substitution:
   If \(t=g(x)\), then
   \(\int f(g(x))g'(x)\,dx=\int f(t)\,dt\)

3. Integration by Parts:
   \(\int u v dx= u\int v dx-\int \Big(\frac{du}{dx}\int v dx\Big)dx\)
   (where \(u,v\) are functions of \(x\)).

4. Partial Fractions:
   Used when integrand is a rational function \(\frac{P(x)}{Q(x)}\) with degree \(P<Q\).

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53.3 Definite Integration

53.3.1 Definition

\[ \int_a^b f(x)\,dx = F(b)-F(a) \] where \(F(x)\) is any antiderivative of \(f(x)\).

This represents the net area under the curve \(y=f(x)\) between \(x=a\) and \(x=b\).

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53.3.2 Properties

1. \(\int_a^a f(x)dx=0\)
   
2. \(\int_a^b f(x)dx=-\int_b^a f(x)dx\)
   
3. \(\int_a^b [f(x)+g(x)]dx=\int_a^b f(x)dx+\int_a^b g(x)dx\)
   
4. \(\int_a^b f(x)dx=\int_a^b f(a+b-x)dx\)

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53.4 Applications of Integration

1. Area under curve:
   \(A=\int_a^b f(x)dx\)
   
2. Area between two curves:
   \(A=\int_a^b [f(x)-g(x)]dx\) where \(f(x)\ge g(x)\) in \([a,b]\).
   
3. Displacement from velocity:
   \(s=\int v(t)dt\).
   
4. Work done:
   \(W=\int F(x)dx\).
   
5. Probability density functions in statistics.

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53.5 Solved Examples

Example 1
Evaluate \(\int (3x^2+2x+1)dx\).
= \(x^3+x^2+x+C\).

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Example 2
Find \(\int \sin^2x dx\).
Use identity: \(\sin^2x=\tfrac{1-\cos 2x}{2}\).
So \(\int \sin^2x dx=\tfrac{x}{2}-\tfrac{\sin 2x}{4}+C\).

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Example 3
Evaluate \(\int_0^1 (x^2+1)dx\).
= \(\Big[\tfrac{x^3}{3}+x\Big]_0^1=\tfrac{1}{3}+1= \tfrac{4}{3}\).

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Example 4
Find \(\int x e^x dx\).
Use parts: \(u=x\), \(dv=e^x dx\).
= \(x e^x-\int e^x dx=xe^x-e^x+C\).

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Example 5
Find area under \(y=x^2\) between \(x=0\) and \(x=2\).
\(A=\int_0^2 x^2 dx=\Big[\tfrac{x^3}{3}\Big]_0^2=\tfrac{8}{3}\).

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Example 6
Evaluate \(\int_0^\pi \sin x dx\).
= \([-\cos x]_0^\pi = (-(-1))-( - (1))=2\).

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53.6 Practice Problems

1. Evaluate \(\int (4x^3-5x+6)dx\).
   
2. Find \(\int \cos^2x dx\).
   
3. Compute \(\int_1^2 (2x+1)dx\).
   
4. Evaluate \(\int e^{2x}dx\).
   
5. Find \(\int \frac{1}{x^2+1}dx\).
   
6. Evaluate \(\int_0^{\pi/2} \sin^2x dx\).
   
7. Find \(\int \frac{x}{x^2+1}dx\).
   
8. Find area under \(y=3x+2\) between \(x=0\) and \(x=4\).
   
9. Solve \(\int x^2 e^x dx\).
   
10. Find \(\int_0^1 \frac{1}{\sqrt{1-x^2}} dx\).

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53.7 Answer Key (Concise)

1. \(x^4-2.5x^2+6x+C\)
   
2. \(\tfrac{x}{2}+\tfrac{\sin 2x}{4}+C\)
   
3. \([x^2+x]_1^2=6\)
   
4. \(\tfrac{1}{2}e^{2x}+C\)
   
5. \(\tan^{-1}x+C\)
   
6. \(\pi/4\)
   
7. \(\tfrac{1}{2}\ln(1+x^2)+C\)
   
8. \(\int_0^4 (3x+2)dx=[1.5x^2+2x]_0^4=40\)
   
9. Use parts: \(=x^2 e^x-2\int x e^x dx=(x^2-2x+2)e^x+C\)
   
10. \(\sin^{-1}(x)\Big|_0^1=\pi/2\)

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53.8 Summary

• Integration is the reverse of differentiation.
  
• Indefinite integrals include a constant \(C\); definite integrals give a numerical value.
  
• Key techniques: substitution, parts, partial fractions.
  
• Applications include area, volume, work, and statistics.
  
• Mastery of standard integrals and properties of definite integrals is essential for exam success.