INTEGRALS

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Top Concepts

1.Integration is the inverse process of differentiation. The process of finding the function from its primitive is known as integration or antidifferentiation.

2.The problem of finding a function whenever its derivative is given leads to indefinite form of integrals.

3.The problem of finding the area bounded by the graph of a function under certain conditions leads to a definite form of integrals.

4.Indefinite and definite integrals together constitute Integral Calculus.

5.Indefinite integral

\int f (x) d x = F (x) + C,

where F(x) is the antiderivative of f(x).

6.Functions with same derivatives differ by a constant.

\int f (x) d x

means integral of f with respect to x, f(x) is the integrand, x is the variable of integration and C is the constant of integration.

8.Geometrically indefinite integral is the collection of family of curves, each of which can be obtained by translating one of the curves parallel to itself.

Family of curves representing the integral of 3x²

$\int f (x) d x = F (x) + C,$ represents a family of curves where different values of C correspond to different members of the family, and these members are obtained by shifting any one of the curves parallel to itself.

9.Properties of antiderivatives

10.Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent.

11.Comparison between differentiation and integration

1.Both are operations on functions.

2.Both satisfy the property of linearity.

3.All functions are not differentiable and all functions are not integrable.

4.The derivative of a function is a unique function, but the integral of a function is not.

5.When a polynomial function P is differentiated, the result is a polynomial whose degree is 1 less than the degree of P. When a polynomial function P is integrated, the result is a polynomial whose degree is 1 more than that of P.

6.The derivative is defined at a point P and the integral of a function is defined over an interval.

7.Geometrical meaning: The derivative of a function represents the slope of the tangent to the corresponding curve at a point. The indefinite integral of a function represents a family of curves placed parallel to each other having parallel tangents at the points of intersection of the family with the lines perpendicular to the axis.

8.The derivative is used for finding some physical quantities such as the velocity of a moving particle when the distance traversed at any time t is known. Similarly, the integral is used in calculating the distance traversed when the velocity at time I is known.

9.Differentiation and integration, both are processes involving limits.

10.By knowing one antiderivative of function f, an infinite number of antiderivatives can be obtained.

12.Integration can be done by using many methods. Prominent among them are

1.Integration by substitution

2.Integration using partial fractions

3.Integration by parts

4.Integration using trigonometric identities.

13.A change in the variable of integration often reduces an integral to one of the fundamental integrals. Some standard substitutions are

14.A function of the form

\frac{P (x)}{Q (x)}

is known as a rational function. Rational functions can be integrated using partial fractions.

15.Partial fraction decomposition or partial fraction expansion is used to reduce the degree of either the numerator or the denominator of a rational function.

16.Integration using partial fractions

A rational function $\frac{P (x)}{Q (x)}$ can be expressed as the sum of partial fractions if $\frac{P (x)}{Q (x)} .$ This takes any of the forms:

17.To find the integral of the product of two functions, integration by parts is used. I and II functions are chosen using the ILATE rule:

I – inverse trigonometric

L – logarithmic

A – algebraic

T – trigonometric

E – exponential is used to identify the first function.

18.Integration by parts

Integral of the product of two functions = (first function) × (integral of the second function) – integral of [(differential coefficient of the first function) × (integral of the second function)].

19.Definite integral

\int_{a}^{b} . f (x) d x

of the function f(x) from limits a to b represents the area enclosed by the graph of the function f(x), the x-axis and the vertical markers x = ‘a’ and x = ‘b’.

20.Definite integral as the limit of a sum: The process of evaluating a definite integral by using the definition is called integration as the limit of a sum or integration from first principles.

21.Method of evaluating

\int_{a}^{b} f (x) d x

i.Calculate antiderivative F(x)

ii.Calculate F(b ) – F(a)

22.Area function

23.Fundamental Theorem of Integral Calculus

●First fundamental theorem of integral calculus: If area function,

A (x) = \int_{a}^{x} f (x) d x

for all x

\geq

a, and f is continuous on [a, b]. Then A’(x) = f(x) for all x

\in

[a, b]

●Second fundamental theorem of integral calculus: Let f be a continuous function of x in the closed interval [a, b] and let F be antiderivative of

\frac{d}{d x} f (x) = f (x)

for all x in domain of f, then

Top Formulae

1.Some Standard Integrals

2.Integral of some special functions

3.Integration by parts

4.Integral as the limit of sums:

5.Different methods of integration

2.Evaluation of integrals of the form

\int . (a x + b) \sqrt{c x + d d x}

Step:

3.Evaluation of integrals of the form

\int . \frac{(a x + b)}{\sqrt{c x + d}} d x

Step:

i.Represent (ax + b) in terms of (cx + d) as follows:

(ax + b) = A (cx + d) + B

ii.Find A and B by equating coefficients of like powers of x on both sides

iii.Replace (ax + b) by A(cx + d) + B in the given integral to obtain

4.Evaluation of integrals of the form

\int . x d x, \int . {c o s}^{m} x d x, w h e r e m \leq 4

Let us express sin^m x and cos^mx in terms of sines and cosines of multiples of x by using the following identities:

5.Evaluation of integrals of the form

$\int s i n m x . c o s n x d x, \int s i n m x . s i n n x d x, \int c o s m x . c o s n x d x$

Let us use the following identities:

6.Evaluation of integrals of the form

\int . \frac{f^{‘} (x)}{f (x)} d x

7.Evaluation of integrals of the form

8.Evaluation of integrals of the form

9.Evaluation of integrals of the form

10.Evaluation of integrals of the form

\int x . {c o s}^{n} x d x, w h e r e m, n \in N

Steps:

i.Check the exponents of sinx and cosx

ii.If the exponent of sinx is an odd positive integer, then put cosx = v

If the exponent of cosx is an odd positive integer, then put sinx = v

If the exponents of both sinx and cosx are odd positive integers, then put either sinx = v or cosx = v

If the exponents of both sinx and cosx are even positive integers, then rewrite sin^m x cosⁿ x in terms of sines and cosines of multiples of x by using trigonometric results.

iii.Evaluate the integral in step (ii)

11.Evaluation of integrals of the form

12.Evaluation of integrals of the form

\int . \frac{d x}{a x^{2} + b x + c}

Steps:

i.Multiply and divide the integrand by x² and make the coefficient of x² unity

ii.Observe the coefficient of x

iii.Add and subtract

{(\frac{1}{2} c o e f f i c i e n t x)}^{2}

to the expression in the denominator

iv.Express the expression in the denominator in the form

v.Use the appropriate formula to integrate.

13.Evaluation of integrals of the form

\int . \frac{d x}{a x^{2} + b x + c}

Steps:

i.Multiply and divide the integrand by x² and make the coefficient of x² unity

ii.Observe the coefficient of x

iii.Add and subtract

{(\frac{1}{2} c o e f f i c i e n t x)}^{2}

inside the square root

iv.Express the expression inside the square root in the form Diagram Description automatically generated

v.Use the appropriate formula to integrate.

14.Evaluation of integrals of the form

\int \frac{p x + q}{a x^{2} + b x + c} d x

Steps:

i.Rewrite the numerator as follows:

15.Evaluation of integrals of the form

\int \frac{p x + q}{a x^{2} + b x + c} d x

where p(x) is a polynomial degree greater than or equal to 2

Steps:

i.Divide the numerator by the denominator, and rewrite the integrand as

16.Evaluation of integrals of the form

\int . \frac{p x + q}{\sqrt{a x^{2} + b x + c}} d x

Steps:

17.Evaluation of integrals of the form

Steps:

i.Divide the numerator and denominator by cos²x

ii.In the denominator, replace sec² x by1 + tan² x

iii.Substitute tan x = v; sec²xdx = dv

iv.Apply the appropriate method to integrate the integral

\int . \frac{d v}{a v^{2} + b v + c}

18.Evaluation of integrals of the form

19.Alternate method: Evaluation of integrals of the form

20.Evaluation of integrals of the form

21.Evaluation of integrals of the form

22.Evaluation of integrals of the form

23.Evaluation of the integrals of the form

\int \sqrt{a x^{2} + b x + c d x}

24.Evaluation of the integrals of the form

\int (m x + n) \sqrt{a x^{2} + b x + c d x}

25.Evaluation of the integrals of the form

26.Evaluation of integration of irrational algebraic functions,

27.Evaluation of integration of irrational algebraic functions,

28.Evaluation of integration of irrational algebraic functions,

29.Evaluation of integration of irrational algebraic functions,

6.Properties of definite integrals

Important Questions

Multiple Choice questions-

1.The anti-derivative of (√x +

\frac{1}{\sqrt x}

) equals

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2. If $\frac{1}{d x}$ (f(x)) = 4x³ – $\frac{3}{x^{4}}$ such that f(2) = 0 then f(x) is ……………

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(a) 10^x – x¹⁰ + c

(b) 10^x + x¹⁰ + c

(d) log (10^x + x¹⁰) + c.

(a) tan x + cot x + c

(b) tan x – cot x + c

(d) tan x – cot 2x + c.

(a) tan x + cot x + c

(b) tan x + cosec x + c

(d) tan x + sec x + c.

(a) -cot (xe^x) + c

(b) tan (xe^x) + c

(d) cot (e^x) + c

(a) x tan^-1 (x + 1) + c

(b) tan^-1 (x + 1) + c

(d) tan^-1 x + c.

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10.

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Very Short Questions:

1.Find ∫

\frac{3 + 3 c o s x}{x + s i n x} d x

(C.B.S.E. Sample Paper 2019-20)

2.Find: ∫ (cos² 2x – sin² 2x)dx. (C.B.S.E. Sample Paper 2019-20)

3.Find: ∫

\frac{d x}{\sqrt{5} - 4 x - 2 x^{2}}

(C.B.S.E. Outside Delhi 2019)

4.Evaluate ∫

\frac{x^{3} - 1}{x^{2}}

dx (N.C.E.R.T. C.B.S.E. 2010C)

5.Find: ∫

\frac{{s i n}^{2} x - {c o s}^{2} x}{s i n x c o s x} d x

(A.I.C.B.S.E. 2017)

6.Write the value of ∫

\frac{d x}{x^{2} + 16}

7.Evaluate: ∫ (x³ + 1) dx. (C.B.S.E. Sample Paper 2019-20)

8.Evaluate:

\int_{0}^{π / 2} e^{x}

(sin x – cos x) dx. (C.B.S.E. 2014)

9.Evaluate:

\int_{0}^{2} \sqrt{4} - x^{2} d x

(A.I.C.B.S.E. 2014)

10.Evaluate: If f(x) =

\int_{0}^{x} t

sin t dt, then write the value of f’ (x). (A.I. C.B.S.E. 2014)

Short Questions:

1.Evaluate:

2.Find: ∫

\frac{{s e c}^{2} x}{\sqrt{{t a n}^{2} x + 4}}

3.Find: ∫

\sqrt{1 - s i n 2 x d x,} \frac{π}{4}

< x <

\frac{π}{2}

4.Find ∫ sinx . log cos x dx (C.B.S.E 2019 C)

5.Find: ∫

\frac{(x^{2} + {s i n}^{2} x) {s e c}^{2} x}{1 + x^{2}}

dx (CBSE Sample Paper 2018-19)

6.Evaluate ∫

\frac{e^{x} (x - 3)}{{(x - 1)}^{3}}

dx (CBSE Sample Paper 2018-19)

7.Find ∫sin^-1 (2x) dx

8.Evaluate:

\int_{- π}^{π} (1 - x^{2})

sin x cos2 x dx.

Long Questions:

1.Evaluate: ∫

\frac{{s i n}^{6} x + {c o s}^{6} x}{{s i n}^{2} x {c o s}^{2} x}

dx (C.B.S.E. 2019 (Delhi))

2.Integrate the function

\frac{c o s (x + a)}{s i n (x + b)}

w.r.t. x. (C.B.S.E. 2019 (Delhi))

3.Evaluate: ∫ x² tan^-1 x dx. (C.B.S.E. (F) 2012)

4.Find: ∫ [log (log x) +

\frac{1}{{(l o g x)}^{2}}

] dx (N.C.E.R.T.; A.I.C.B.S.E. 2010 C)

Case Study Questions-

1. Integration is the process of finding the antiderivative of a function. In this process, we are provided with the derivative of a function and asked to find out the function (i.e., Primitive) Integration is the inverse process of differentiation.

Let f (x) be a function of x. If there is a function g(x), such that d/dx (g(x)) = f (x), then g(x) is called an integral of f (x) w.r.t x and is denoted by ∫f (x )dx = g(x) + c, where c is constant of integration.

(i) ∫(3x+4)³ dx is equal to:

(ii)

(iii)∫sin²(x) dx is equal to:

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(iv) ∫tan²(x) dx is equal to:

(v)

2. When the integrated can be expressed as a product of two functions, one of which can be differentiated and the other can be integrated, then we apply integration by parts. If f(x) = first function (that can be differentiated) and g(x) = second function (that can be integrated), then the preference of this order can be decided by the word “ILATE”, where

I stands for Inverse Trigonometric Function
L stands for Logarithmic Function
A stands for Algebraic Function
T stands for Trigonometric Function
E stands for Exponential Function, then

(i) ∫x.sin3x dx =

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(ii) ∫ log(x + 1) dx =

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(iii)

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(iv)

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Answer Key-

Multiple Choice questions-

3.Answer: (d) log (10^x + x¹⁰) + c.

4.Answer: (b) tan x – cot x + c

5.Answer: (a) tan x + cot x + c

6.Answer: (b) tan (xe^x) + c

Answer: (b) tan^-1 (x + 1) + c

10.

Very Short Answer:

1.Solution:

I = ∫ $\frac{3 + 3 c o s x}{x + s i n x}$ dx = 3 log lx + sin xl + c.

[∵ Num. = $\frac{d}{d x}$ denom.]

2.Solution:

I = ∫cos 4x dx = $\frac{s i n 4 x}{4}$ + c

3.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 1

4.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 2

5.Solution:

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6.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 4

7.Solution:

I = $\int_{- 2}^{2} x^{3}$ dx + $\int_{- 2}^{2} 1$ ⋅dx = I₁

⇒ 2 – (-2) = 4.

8.Solution:

$\int_{0}^{π / 2} e^{x}$ (sin x – cos x)dx

$\int_{0}^{π / 2} e^{x}$ (-cos x + sin x)dx

I” Form: ∫e^x (f(x) + f'(x) dx”

= -e ^π/2cos $\frac{π}{2}$ + e⁰ cos 0

= -e ^π/2 (0) + (1) (1)

= -0 + 1 = 1

9.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 5

= [0 + 2 sin^-1(1)] – [0 + 0]

= 2sin^-1(1) = 2(π/2) = π

10.Solution:

We have: f(x) = $\int_{0}^{x} t$ sin t dt.

f'(x) = x sin x. $\frac{d}{d x}$ (x) – 0

[Property XII; Leibnitz’s Rule]

= x sin x . (1)

= x sin x.

Short Answer:

1.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 6

2.Solution:

I = $\frac{{s e c}^{2} x}{\sqrt{{t a n}^{2} x + 4}}$

Put tan x = t so that sec² x dx = dt.

∴ I = ∫ $\frac{d t}{\sqrt{t^{2} + 2^{2}}}$

= log |t + $\sqrt{t^{2} + 4}$ | + C

= log |tan x + $\sqrt{t^{2} + 4}$ | + C

3.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 7

Class 12 Maths Important Questions Chapter 7 Integrals 7 - 1

4.Solution:

∫sinx . log cos x dx

Put cox x = t

so that – sin x dx = dt

i.e., sin x dx = – dt.

∴ I = –∫log t.1dt

= -[ log t.t – ∫ 1/t. t dt ]

[Integrating by parts]

= – [t log t – t] + C = f(1 – log t) + C

= cos x (1 – log (cos x)) + C.

5.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 7 - 2

6.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 7 - 3

7.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 7 - 4

8.Solution:

Here, f(x) = (1 – x²) sin x cos² x.

f(x) = (1 – x²) sin (-x) cos² (-x)

= – (1 – x²) sin x cos² x

= -f(x)

⇒ f is an odd function.

Hence, I = 0.

Long Answer:

1.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 14

Class 12 Maths Important Questions Chapter 7 Integrals 15

2.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 16

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3.Solution:

Class 12 Maths Important Questions Chapter 7 Integrals 18

4.Solution:

Let ∫ [log (log x) + $\frac{1}{{(l o g x)}^{2}}$ ] dx

= ∫ log (log x) dx + ∫ $\frac{1}{{(l o g x)}^{2}}$ ] dx …… (1)

Let I = I₁ + I₂

Now I₁ = ∫ log (log x) dx

=∫ log (log x) 1 dx

= log (log x).x – ∫ $\frac{1}{l o g x \cdot x}$ x.dx

(Integrating by parts)

= xlog(logx) – ∫ $\frac{1}{l o g x}$ dx ……….. (2)

Let I₁ = I₃ + I₄

Class 12 Maths Important Questions Chapter 7 Integrals 19

Putting in (2),

Putting in (1),

I = x log (log x)

Class 12 Maths Important Questions Chapter 7 Integrals 20

Case Study Answers-

(i)

(ii)

(iii)

(iv) (d) tan x – x + c

(v)

(i)

(ii)

(iii)

(iv)

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