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01 • INTRODUCTION

What is Integration?

Integration is one of the two fundamental processes of calculus. It is closely connected with differentiation and can be understood as the inverse process of differentiation.

If the derivative of a function F(x) is f(x), then an antiderivative of f(x) is F(x).

F′(x) = f(x)

⇒ ∫ f(x) dx = F(x) + C

Here C is an arbitrary constant called the constant of integration. The symbol ∫ is called the integral sign.

02 • CONCEPT

Integration as the Inverse Process of Differentiation

Suppose F′(x) = f(x). Then all functions whose derivative is f(x) differ only by a constant. Hence,

∫ f(x) dx = F(x) + C

Example 1

∫ 3x² dx = x³ + C

Example 2

∫ cos x dx = sin x + C

Example 3

∫ 1/x dx = ln|x| + C

03 • FORMULAS

Standard Integrals

No.	Integral	Result
1	∫ xⁿ dx	xⁿ⁺¹/(n+1) + C, n ≠ −1
2	∫ dx/x	ln\|x\| + C
3	∫ eˣ dx	eˣ + C
4	∫ aˣ dx	aˣ / ln a + C
5	∫ sin x dx	−cos x + C
6	∫ cos x dx	sin x + C
7	∫ sec²x dx	tan x + C
8	∫ csc²x dx	−cot x + C
9	∫ sec x tan x dx	sec x + C
10	∫ csc x cot x dx	−csc x + C
11	∫ sec x dx	ln\|sec x + tan x\| + C
12	∫ csc x dx	ln\|csc x − cot x\| + C
13	∫ tan x dx	ln\|sec x\| + C
14	∫ cot x dx	ln\|sin x\| + C
15	∫ dx/(x²+a²)	(1/a) tan⁻¹(x/a) + C
16	∫ dx/√(a²−x²)	sin⁻¹(x/a) + C
17	∫ dx/(a²−x²)	(1/2a) ln\|(a+x)/(a−x)\| + C
18	∫ dx/(x²−a²)	(1/2a) ln\|(x−a)/(x+a)\| + C

04 • METHOD

Integration by Substitution

Substitution is useful when the integrand contains a function and its derivative, or can be transformed into a simpler standard integral.

Put u = g(x)

du = g′(x) dx

∫ f(g(x))g′(x) dx = ∫ f(u) du

Examples

Example 1

∫ 2x cos(x²) dx

Put u = x²

du = 2x dx

Therefore, integral = ∫ cos u du

= sin u + C = sin(x²) + C

Example 2

∫ x/(x²+4) dx

Put u = x² + 4

du = 2x dx

∴ x dx = du/2

∴ Integral = 1/2 ln(x²+4) + C

05 • METHOD

Integration by Parts

∫ u dv = uv − ∫ v du

Integration by parts is particularly useful for integrals involving products of two functions.

Choosing u: ILATE / LIATE

I

Inverse Trigonometric

L

Logarithmic

A

Algebraic

T

Trigonometric

E

Exponential

Example: Evaluate ∫ x eˣ dx

Let u = x, dv = eˣ dx

du = dx, v = eˣ

∫x eˣ dx = x eˣ − ∫eˣ dx

= eˣ(x − 1) + C

Example: Evaluate ∫ ln x dx

Let u = ln x, dv = dx

du = dx/x, v = x

∫ln x dx = x ln x − ∫x(1/x)dx

= x ln x − x + C

06 • IMPORTANT METHOD

Partial Fractions — Detailed Notes

Partial fraction decomposition is a method of rewriting a rational function as a sum of simpler rational functions. It is especially useful when the denominator can be factorised into linear or quadratic factors.

Rational Function =

Sum of Simpler Fractions

Step-by-Step Procedure

1

Check the Degree

If degree of numerator is greater than or equal to degree of denominator, first perform polynomial division.

2

Factorise the Denominator

Factor the denominator into linear and/or irreducible quadratic factors.

3

Write the Correct Form

Use the appropriate partial fraction form according to the factors.

4

Find the Constants

Multiply by the denominator and compare coefficients or substitute convenient values.

Case I: Distinct Linear Factors

If denominator = (x−a)(x−b),

P(x)/[(x−a)(x−b)] = A/(x−a) + B/(x−b)

Example

Evaluate ∫ dx/(x²−1)

x²−1 = (x−1)(x+1)

1/[(x−1)(x+1)] = A/(x−1) + B/(x+1)

1 = A(x+1) + B(x−1)

Put x = 1 ⇒ A = 1/2

Put x = −1 ⇒ B = −1/2

∴ ∫dx/(x²−1) = 1/2 ln|x−1| − 1/2 ln|x+1| + C

= 1/2 ln |(x−1)/(x+1)| + C

Case II: Repeated Linear Factors

For (x−a)²:

A/(x−a) + B/(x−a)²

For (x−a)³:

A/(x−a) + B/(x−a)² + C/(x−a)³

Example

1/[x(x−1)²]

= A/x + B/(x−1) + C/(x−1)²

The repeated factor (x−1)² requires both powers to be included.

Case III: Irreducible Quadratic Factor

If Q(x) is an irreducible quadratic factor, use a linear numerator:

(Ax+B)/(x²+px+q)

The numerator must generally have degree one less than the degree of the irreducible quadratic denominator.

Case IV: Product of Linear and Quadratic Factors

For (x−a)(x²+px+q),

A/(x−a) + (Bx+C)/(x²+px+q)

Important Solved Example

Evaluate: ∫ (3x+5)/(x²+3x+2) dx

x² + 3x + 2 = (x+1)(x+2)

Therefore,

(3x+5)/[(x+1)(x+2)] = A/(x+1) + B/(x+2)

3x+5 = A(x+2) + B(x+1)

Put x = −1: 2 = A

Put x = −2: −1 = −B ⇒ B = 1

Hence,

Integral = ∫[2/(x+1) + 1/(x+2)]dx

= 2ln|x+1| + ln|x+2| + C

Partial Fraction Checklist

✓ Check numerator and denominator degrees.

✓ Factorise the denominator completely.

✓ Identify repeated factors.

✓ Use the correct decomposition form.

✓ Find constants carefully.

✓ Integrate each simpler fraction.

✓ Use absolute values with logarithms where required.

✓ Differentiate the final answer to verify.

07 • DEFINITE INTEGRALS

Definite Integrals

A definite integral has an upper and lower limit. Unlike an indefinite integral, its value is a number for a suitable continuous function.

∫ₐᵇ f(x) dx

Lower limit = a Upper limit = b

Important: Signed Area

A definite integral represents signed accumulation. The part of the graph above the x-axis contributes positively, while a part below the x-axis contributes negatively.

08 • THEOREM

Fundamental Theorem of Calculus

PART I

Differentiating an Integral Function

F(x) = ∫ₐˣ f(t)dt

F′(x) = f(x)

If f is continuous, the derivative of the accumulation function is the original function.

PART II

Evaluation Rule

F′(x) = f(x)

∫ₐᵇ f(x)dx = F(b) − F(a)

This rule converts the evaluation of a definite integral into evaluation of an antiderivative.

09 • METHOD

Definite Integral by Substitution

∫ₐᵇ f(g(x))g′(x)dx = ∫_g(a)^g(b) f(u)du

When substitution is used directly in a definite integral, the limits should also be changed to the corresponding values of the new variable.

Example

I = ∫₀¹ 2x/(1+x²) dx

Put u = 1+x²

du = 2x dx

When x=0, u=1

When x=1, u=2

I = ∫₁² du/u

I = [ln u]₁² = ln 2

10 • MOST IMPORTANT

Properties of Definite Integrals — In Detail

These properties are extremely useful in CBSE-style questions because many definite integrals can be simplified without directly finding an antiderivative.

1

Same Upper and Lower Limits

∫ₐᵃ f(x) dx = 0

There is no interval when the upper and lower limits are the same. Hence the accumulated value is zero.

Example: ∫₃³ (x²+1)dx = 0

2

Reversal of Limits

∫ₐᵇ f(x)dx = −∫ᵇₐ f(x)dx

Interchanging the upper and lower limits changes the sign of the integral.

Example: ∫₂⁵ x dx = −∫₅² x dx

3

Additivity Over an Interval

∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx

Any point c between a and b may be used to split the integral into two simpler integrals.

∫₀³ f(x)dx = ∫₀¹ f(x)dx + ∫₁³ f(x)dx

4

Constant Multiple Property

∫ₐᵇ kf(x)dx = k∫ₐᵇ f(x)dx

A constant multiplier may be taken outside a definite integral.

5

Linearity

∫ₐᵇ [f(x) ± g(x)]dx = ∫ₐᵇ f(x)dx ± ∫ₐᵇ g(x)dx

A sum or difference can be separated into individual definite integrals.

6

Comparison Property

If f(x) ≥ g(x) on [a,b],

then ∫ₐᵇ f(x)dx ≥ ∫ₐᵇ g(x)dx

This property follows from the fact that the integral preserves inequalities for continuous functions.

7

Non-Negative Function

If f(x) ≥ 0 on [a,b],

then ∫ₐᵇ f(x)dx ≥ 0

When the graph lies on or above the x-axis, the definite integral cannot be negative.

8

Reflection / Substitution Property

∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx

This is one of the most useful properties for CBSE problems. It follows by making the substitution x = a+b−t.

Let I = ∫ₐᵇ f(x)dx

Put x = a+b−t

dx = −dt

x=a ⇒ t=b

x=b ⇒ t=a

Hence I = ∫ₐᵇ f(a+b−x)dx

9

Integral of an Even Function

If f(−x) = f(x),

∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx

The two halves of the graph are mirror images about the y-axis and therefore contribute equally.

Example: ∫₋₂² x⁴ dx = 2∫₀² x⁴ dx = 64/5

10

Integral of an Odd Function

If f(−x) = −f(x),

∫₋ₐᵃ f(x)dx = 0

The areas on the two sides of the y-axis cancel algebraically.

Example: ∫₋₂² x³ dx = 0

11

Periodic Function Property

If f(x) has period T,

∫ₐᵃ⁺ᵀ f(x)dx = ∫₀ᵀ f(x)dx

The integral over any interval of length equal to one complete period has the same value.

12

Average Value of a Function

Average Value = 1/(b−a) ∫ₐᵇ f(x)dx

For a continuous function on [a,b], its average value over the interval is obtained by dividing the definite integral by the length of the interval.

13

Product / Sum Useful Transformation

∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx

Therefore,

2I = ∫ₐᵇ [f(x)+f(a+b−x)]dx

This technique is especially useful when adding an integral to its reflected form produces a constant or simpler expression.

14

Bounds for a Definite Integral

If m ≤ f(x) ≤ M on [a,b],

m(b−a) ≤ ∫ₐᵇ f(x)dx ≤ M(b−a)

This gives an upper and lower bound for the value of a definite integral.

Even and Odd Function Visual

Reflection Property: x → a+b−x

Example: Even Function

I = ∫₋₂² x⁴ dx

x⁴ is even.

I = 2∫₀² x⁴ dx

= 2[x⁵/5]₀² = 64/5

Example: Odd Function

I = ∫₋₂² x³ dx

x³ is odd.

Therefore I = 0

High-Value CBSE Technique

Evaluate: I = ∫₀π x sin x dx

By reflection property,

I = ∫₀π (π−x) sin(π−x) dx

Since sin(π−x)=sin x,

I = ∫₀π (π−x)sin x dx

Adding the two expressions,

2I = ∫₀π [x+(π−x)]sin x dx

2I = π∫₀π sin x dx

= π[−cos x]₀π = 2π

∴ I = π

11 • SOLVED EXAMPLES

Important Solved Examples

Example 1: Evaluate ∫(3x²−4x+5)dx

= ∫3x²dx − ∫4xdx + ∫5dx

= x³ − 2x² + 5x + C

Example 2: Evaluate ∫dx/(x²+4)

∫dx/(x²+2²)

= 1/2 tan⁻¹(x/2) + C

Example 3: Evaluate ∫2x/(x²+1) dx

Put u=x²+1

du=2x dx

Answer = ln(x²+1)+C

Example 4: Evaluate ∫xeˣdx

u=x, dv=eˣdx

du=dx, v=eˣ

= xeˣ − eˣ + C

= eˣ(x−1)+C

Example 5: Evaluate ∫ln x dx

u=lnx, dv=dx

du=dx/x, v=x

= xlnx − ∫dx

= xlnx − x + C

Example 6: Partial Fractions — ∫dx/[x(x+1)]

1/[x(x+1)] = 1/x − 1/(x+1)

Therefore,

Answer = ln|x| − ln|x+1| + C

Example 7: Evaluate ∫₀¹(x²+2x)dx

= [x³/3+x²]₀¹

= 1/3 + 1 = 4/3

Example 8: Evaluate ∫₀π/2 sin x dx

= [−cos x]₀π/2

= 1

Example 9: Evaluate ∫₀¹ 2x/(1+x²) dx

Put u=1+x²

Limits: x=0 ⇒ u=1; x=1 ⇒ u=2

I=∫₁²du/u

I=ln2

Example 10: Evaluate ∫₀¹xeˣdx

∫xeˣdx=eˣ(x−1)

I=[eˣ(x−1)]₀¹

=0−(−1)

=1

Example 11: Use even-function property to evaluate ∫₋₂²x⁴dx

x⁴ is even.

I=2∫₀²x⁴dx

=2[x⁵/5]₀²

=64/5

Example 12: Use odd-function property to evaluate ∫₋₂²x³dx

x³ is odd.

Therefore, integral = 0

Example 13: Evaluate ∫₀π x sin x dx using symmetry

I=∫₀πx sinx dx

I=∫₀π(π−x)sin(π−x)dx

sin(π−x)=sinx

2I=π∫₀πsinx dx

2I=2π

I=π

Example 14: Evaluate ∫₀ᵃ dx/√(a²−x²), a>0

Standard result: ∫dx/√(a²−x²)=sin⁻¹(x/a)

I=[sin⁻¹(x/a)]₀ᵃ

I=π/2

Example 15: Find derivative of F(x)=∫₁ˣ(t²+1)dt

By Fundamental Theorem of Calculus,

F′(x)=x²+1

Example 16: Evaluate ∫₀²(3x²+2x)dx

= [x³+x²]₀²

= 8+4=12

12 • PRACTICE

Practice Questions

1. Evaluate ∫(4x³−3x²+2)dx.

x⁴ − x³ + 2x + C

2. Evaluate ∫dx/(x²+9).

(1/3)tan⁻¹(x/3)+C

3. Evaluate ∫2x/(x²+5)dx.

ln(x²+5)+C

4. Evaluate ∫2x cos(x²)dx.

sin(x²)+C

5. Evaluate ∫xeˣdx.

eˣ(x−1)+C

6. Evaluate ∫ln x dx.

xlnx−x+C

7. Resolve 1/(x²−1) into partial fractions.

1/2[1/(x−1)−1/(x+1)]

8. Resolve 1/[x(x+1)] into partial fractions.

1/x−1/(x+1)

9. Evaluate ∫₀¹x²dx.

1/3

10. Evaluate ∫₀π/2cos x dx.

1

11. Evaluate ∫₋₃³x⁵dx.

0

12. Evaluate ∫₋₃³x⁶dx using symmetry.

2×3⁷/7 = 4374/7

13. Evaluate ∫₀¹2x/(1+x²)dx.

ln2

14. Evaluate ∫₀¹xeˣdx.

1

15. Evaluate ∫₀πx sin x dx.

π

16. State the even-function property.

If f(−x)=f(x), then ∫₋ₐᵃf(x)dx=2∫₀ᵃf(x)dx.

17. State the odd-function property.

If f(−x)=−f(x), then ∫₋ₐᵃf(x)dx=0.

18. State the reflection property.

∫ₐᵇf(x)dx=∫ₐᵇf(a+b−x)dx.

19. Find F′(x) if F(x)=∫₂ˣ(t³+2t)dt.

F′(x)=x³+2x.

20. Evaluate ∫₀ᵃ dx/√(a²−x²), a>0.

π/2

13 • HOT QUESTIONS

Higher Order Thinking Questions

HOT 1. Prove that ∫ₐᵇf(x)dx = ∫ₐᵇf(a+b−x)dx.

Use the substitution x=a+b−t. Then dx=−dt. The limits transform from a,b to b,a and reversing them removes the negative sign.

HOT 2. Evaluate ∫₀π x sin x dx without directly using integration by parts.

Answer = π

Use the reflection property x→π−x.

HOT 3. Show that the integral of an odd function over [−a,a] is zero.

Split the integral at zero and use f(−x)=−f(x). The two contributions cancel.

HOT 4. Evaluate ∫₀ᵃ f(x)dx + ∫₀ᵃ f(a−x)dx.

= 2∫₀ᵃf(x)dx

HOT 5. If m≤f(x)≤M on [a,b], establish bounds for ∫ₐᵇf(x)dx.

m(b−a)≤∫ₐᵇf(x)dx≤M(b−a)

HOT 6. Evaluate a rational integral using partial fractions and logarithmic standard results.

Factor the denominator, write the decomposition, determine constants and integrate each term.

HOT 7. If F(x)=∫₀ˣ(t²+1)dt, find F′(x) and F″(x).

F′(x)=x²+1, F″(x)=2x

HOT 8. Evaluate a definite integral involving an even function without finding its antiderivative.

First test whether f(−x)=f(x). If yes, replace the integral over [−a,a] by twice the integral from 0 to a.

HOT 9. Explain why +C is not written in a definite integral answer.

In F(b)−F(a), the same arbitrary constant occurs at both limits and cancels.

HOT 10. Determine whether a given integrand is suitable for substitution or integration by parts.

Look for a composite function together with its derivative for substitution. For a product of unlike functions, consider integration by parts.

14 • REVISION

Quick Revision Sheet

Concept	Key Result
Indefinite Integral	∫f(x)dx=F(x)+C
Power Rule	∫xⁿdx=xⁿ⁺¹/(n+1)+C
Logarithmic	∫dx/x=ln\|x\|+C
Substitution	u=g(x), du=g′(x)dx
By Parts	∫u dv=uv−∫vdu
Partial Fractions	Factor denominator and decompose.
FTC	∫ₐᵇf(x)dx=F(b)−F(a)
Same Limits	∫ₐᵃf(x)dx=0
Reversed Limits	∫ₐᵇf=−∫ᵇₐf
Additivity	∫ₐᵇf=∫ₐᶜf+∫ᶜᵇf
Reflection	∫ₐᵇf(x)dx=∫ₐᵇf(a+b−x)dx
Even Function	∫₋ₐᵃf=2∫₀ᵃf
Odd Function	∫₋ₐᵃf=0
Average Value	1/(b−a) ∫ₐᵇf(x)dx

15 • EXAM TIPS

CBSE Exam Tips for Integrals

✓

Add +C

Always write the constant of integration in indefinite integrals.

✓

No +C in Definite Integrals

The arbitrary constant cancels when evaluating F(b)−F(a).

✓

Factor First

In partial fractions, factorise the denominator completely before choosing the decomposition.

✓

Repeated Factors Matter

For (x−a)², include terms for both 1/(x−a) and 1/(x−a)².

✓

Look for Symmetry

Before calculating a definite integral over [−a,a], check whether the integrand is even or odd.

✓

Use x → a+b−x

This transformation is powerful when f(x)+f(a+b−x) becomes simpler.

✓

Change Limits Consistently

When substitution is performed in a definite integral, either change the limits or return completely to the original variable.

✓

Check the Sign

Remember that reversing limits changes the sign and that area below the x-axis contributes negatively to the signed integral.

Final Exam Checklist

□ Memorise standard integrals.

□ Practise substitution.

□ Practise integration by parts.

□ Master partial fractions.

□ Learn all major definite-integral properties.

□ Practise even/odd function questions.

□ Practise x → a+b−x transformation.

□ Revise Fundamental Theorem of Calculus.

□ Check limits carefully.

□ Verify signs and constants.

Integrals

What is Integration?

Integration as the Inverse Process of Differentiation

Standard Integrals

Integration by Substitution

Examples

Integration by Parts

Choosing u: ILATE / LIATE

Partial Fractions — Detailed Notes

Step-by-Step Procedure

Check the Degree

Factorise the Denominator

Write the Correct Form

Find the Constants

Case I: Distinct Linear Factors

Case II: Repeated Linear Factors

Case III: Irreducible Quadratic Factor

Case IV: Product of Linear and Quadratic Factors

Important Solved Example

Partial Fraction Checklist

Definite Integrals

Important: Signed Area

Fundamental Theorem of Calculus

Differentiating an Integral Function

Evaluation Rule

Definite Integral by Substitution

Example

Properties of Definite Integrals — In Detail

Same Upper and Lower Limits

Reversal of Limits

Additivity Over an Interval

Constant Multiple Property

Linearity

Comparison Property

Non-Negative Function

Reflection / Substitution Property

Integral of an Even Function

Integral of an Odd Function

Periodic Function Property

Average Value of a Function

Product / Sum Useful Transformation

Bounds for a Definite Integral

Even and Odd Function Visual

Reflection Property: x → a+b−x

Example: Even Function

Example: Odd Function

High-Value CBSE Technique

Important Solved Examples

Practice Questions

Higher Order Thinking Questions

Which Method Should I Use?

Quick Revision Sheet

CBSE Exam Tips for Integrals

Add +C

No +C in Definite Integrals

Factor First

Repeated Factors Matter

Look for Symmetry

Use x → a+b−x

Change Limits Consistently

Check the Sign

Final Exam Checklist

Common Mistakes to Avoid

Master Class XII Mathematics