By going through these CBSE Class 12 Maths Notes Chapter 7 Integrals, students can recall all the concepts quickly.

## Integrals Notes Class 12 Maths Chapter 7

→ Integration is the inverse process of differentiation. If we are given the derivative of a function and we have to find the function whose derivative is given, the process of finding the primitive or the original function is called the integration or anti-differentation.

Let \(\frac{d}{d x}\)[F(x) + c] = F ‘(x) = f(x)

⇒ F(x) + c is the antiderivative or integal of f(x). This may be written as ∫f(x)dx = F(x) + c,

where c is an arbitrary constant called constant of integration.

∫f(x) dx is called indefinite integral.

**Properties of Indefinite Integral.**

- The processes of differentiation and integration are inverse processes of each other, i.e.,

\(\frac{d}{d x}\)∫f(x) dx = f(x). - Indefinite integrals with the same derivatives belong to the same family of curves and so they are equivalent, i.e.,

in ∫f(x) dx = F(x) + c, [F(x) + c] denotes the same family of indefinite integrals of f(x). - ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx.
- ∫k f(x) dx = k ∫f(x) dx, where k is real number.
- If k
_{1}, k_{2},………… k_{n}are the real numbers, then ∫[k_{1}f_{1}(x)+ k_{2 }f_{2}(x) +………..+ k_{n}f_{n}(x)] dx = k_{1}∫f_{1}(x)dx +k_{2 }∫f_{2}(x) dx + +k_{n}∫f_{n}(x)dx.

→ We know the formulae for the derivatives of many functions. Corresponding integrals are given below:

Derivatives |
Integrals |

1. \( \frac{d}{d x}\left(\frac{x^{n+1}}{n+1}\right) \) Where n = 0, \(\frac{d}{d x} \)(x) = 1 |
1. ∫x^{n} dx = \( \frac{x^{n+1}}{n+1}\) + C∫dx = x + C |

Trigonometric Functions 2. \(\frac{d}{d x} \)(sin x) = cos x |
2. ∫cos x dx = sin x + C |

3. \(\frac{d}{d x} \)(tan x ) = sin x | 3. ∫sin x dx = – cos x + C |

4. \(\frac{d}{d x} \)(- cosec x) = cosec x cot x | 4. ∫sec^{2} x dx = tan x + C |

5. \(\frac{d}{d x} \)(- cosec x) = cosec x cot x | 5. ∫cosec x cot x dx = – cosec x + C |

6. \(\frac{d}{d x} \)(sec x) = sec x tan x | 6. ∫sec x tan x dx = sec x + C |

7. \(\frac{d}{d x} \)(- cot x) = cosec^{2} x |
7. ∫cosec^{2} x dx = – cot x + C |

Inverse Trigonometric Functions 8. \(\frac{d}{d x} \)(sin ^{-1 }x) = \( \frac{1}{\sqrt{1-x^{2}}} \) |
8. ∫\( \frac{1}{\sqrt{1-x^{2}}} \) dx = sin^{-1 } x + C |

9. \(\frac{d}{d x} \)(- cos^{-1 }x) = + \( \frac{1}{\sqrt{1-x^{2}}} \) |
9. ∫\( \frac{1}{\sqrt{1-x^{2}}} \) dx = – cos^{-1 }x + C |

10. \(\frac{d}{d x} \)(tan^{-1 }x) = \( \frac{1}{1+x^{2}}\) |
10. ∫\( \frac{1}{1+x^{2}}\) dx = tan^{-1 }x + C |

11. \(\frac{d}{d x} \)(- cot^{-1 }x) = \( \frac{1}{1+x^{2}}\) |
11. ∫\( \frac{1}{1+x^{2}}\) dx = – cot^{-1 }x + C |

12. \(\frac{d}{d x} \)(sec^{-1 }x) = \( \frac{1}{x \sqrt{x^{2}-1}}\) |
12. ∫\( \frac{1}{x \sqrt{x^{2}-1}}\) dx = sec^{-1 }x + C |

13. \(\frac{d}{d x} \)(- cosec^{-1 } x) = \( \frac{1}{x \sqrt{x^{2}-1}}\) |
13. ∫\( \frac{1}{x \sqrt{x^{2}-1}}\) dx = – cosec^{-1 }x + C |

Exponential Functions 14. \(\frac{d}{d x} \)e ^{x} = e^{x} |
14. ∫e^{x} dx = e^{x} + C |

15. \( \frac{d}{d x}\left(\frac{a^{x}}{\log a}\right)\) =ax | 15. ∫ax dx = \( \frac{a^{x}}{\log a}\) + C |

Logarithmic Functions 16. \(\frac{d}{d x} \)(log _{e} x) = \(\frac{1}{x}\) |
16. ∫\(\frac{1}{x}\) dx = log_{e} x + C |

17. \(\frac{1}{2}\)(log_{a} x) = \(\frac{1}{x}\)log_{a} e |
17. ∫\(\frac{1}{x}\)log_{a} e dx = log_{a} x + C |

**Geometrical Interpretation of Indefinite Integral:**

∫f(x) dx = F(x) + C = y (say).

y = F(x) + C represents a family of curves. By giving different values to C, we get different members of family. These members can be obtained by shifting any of the curves parallel to itself.

**Comparison between Differentiation and Integration:**

Differentiation |
Integration |

1. It is an operation on function. | 1. It is an operation on function. |

2. \( \begin{aligned}\frac{d}{d x}\left[k_{1} f_{1}(x)+k_{2} f_{2}(x)+\ldots\right.\\\left.+k_{n} f_{n}(x)\right]\end{aligned}\) = k_{1} \(\frac{d}{d x} \)f_{1} (x) + k_{2} \(\frac{d}{d x} \)f_{2}(x) + ……….+ k_{n} \(\frac{d}{d x} \)f_{n} (x) |
2. ∫ k_{1} \(\frac{d}{d x} \)f_{1} (x) + k_{2} \(\frac{d}{d x} \)f_{2} (x) + ……….+ k_{n} \(\frac{d}{d x} \)f_{n} (x) = k_{1}∫f_{2}1(x) dx + k_{2}∫f_{2} (x) dx + ……… + k_{n}∫f_{n} (x) dx |

3. Some functions are not differentiable. | 3. All functions are not integrable. |

4. The derivative of a function, if it exists, is unique. | 4. The integral of a function is not unique. |

5. ¡f a polynomial function of a degree n is differentiated, we obtain a polynomial of degree n – 1. | 5. If a polynomial function of a degree n is integrated, we get a polynomial of degree n + 1. |

6. We can obtain a derivative at a point. | 6. Integral of a function may be obtained over an interval in which f is defined. |

7. Slope of tangent at a point x = x_{1} is f ‘(x_{1}). |
7. Integral of a function represents a family of curves. |

8. If the distance traversed at any time f is known, we can find velocity and acceleration. | 8. When the velocity or acceleration at any time t is known, we can find the distance traversed in time t, |

9. Differentiation is a process involving limits. | 9. Integration too involves limits. |

**Integration by Substitution:**

Let I = ∫f(x) dx. This integral can be transformed into another form by changing the independent variable x to t by putting x=g(t).

∴ \(\frac{dx}{dt} \) = g'(t) or dx = g'(t) dt

∴ I = ∫f[g(t)]g'(t) dt

Note: While making a substitution, it should be kept in mind that the f[g(t)] is in the form of some standard formula, whereas g'(t) is a factor, along with f[g(t)] e.g.

Consider the integral I = ∫x^{2} cos(x^{3} + 2)dx

If we put x^{3} + 2 = t, its derivative 3x^{2} is a factor and cos t can easily be integrated.

**Some Results:**

- ∫tan x = log|sec x| + C
- ∫cot x = log|sin x| + C
- ∫sec x = log|sec x + tan x| + C
- ∫cosec x dx = log|cosec x – cot x| + C

→ Use of Trigonometric Identities:

Use following trigonometric identities for integrating the functions such as sin2x.cos2x, sin3x, cos3x, sin x cos x, etc.

- sin
^{2}x = \(\frac{1-\cos 2 x}{2}\), cos2 x = \(\frac{1+\cos 2 x}{2}\) - sin
^{3}x = \(\frac{3 \sin x-\sin 3 x}{4}\), cos2 x = \(\frac{3 \cos x+\cos 3 x}{4}\) - 2sinA cosB = sin(A + B) + sin(A – B)

2cosA sinB = sin(A + B) – sin(A – B)

2cosA cosB = cos(A + B) + cos(A – B)

2sinA sinB = cos(A – B) – cos(A + B)

**Some more Integrals:**

→ How to integrate when the integral has ax^{2} + bx + c or \(\sqrt{a x^{2}+b x+c}\) in the denominator?

So, ax^{2} + bx + c changes to t^{2} ± k^{2}.

Thus, \(\frac{1}{a x^{2}+b x+c}\) converts into the form \(\frac{1}{t^{2} \pm k^{2}}\) and

\(\frac{1}{\sqrt{a x^{2}+b x+c}}\) converts into the form \(\frac{1}{\sqrt{t^{2} \pm \dot{k}^{2}}}\).

→ To find the integration of type ∫\(\frac{p x+q}{a x^{2}+b x+c}\) dx or ∫\(\frac{p x+q}{\sqrt{a x^{2}+b x+c}}\) dx:

Put px + q = A\(\frac{d}{d x} \)(ax^{2} + bx + c) + B

Compare the two sides and find the value of A and B.

**Partial Fractions of Rational Functions.**

(a) Let \(\frac{\mathrm{P}(x)}{\mathrm{Q}(x)}\) be a rational function, where P(x) and Q(x) are

polynomials, where Q(x) ≠ 0.

- If degree of P(x) is less than degree of Q(x), then \(\frac{\mathrm{P}(x)}{\mathrm{Q}(x)}\) called a proper rational function.
- If degree of P(x) is greater than the degree of Q(x), then \(\frac{\mathrm{P}(x)}{\mathrm{Q}(x)}\) is known as an improper rational function.

This can be expressed in the form f(x) + \(\frac{\mathrm{P}_{1}(x)}{\mathrm{Q}(x)}\) by dividing P(x) by Q(x), so that degree of P1(x) is less than the degree of Q(x).

(b) 1. Let \(\frac{\mathrm{P}(x)}{\mathrm{Q}(x)}\) be a proper rational function. Find the factors of Q(x). Let these factors be linear.

P(x)

Suppose Q(x) = (x + a)(x + b)(x + c), Then, \(\frac{\mathrm{P}(x)}{\mathrm{Q}(x)}\) is written as

\(\frac{\mathrm{P}(x)}{\mathrm{Q}(x)}=\frac{\mathrm{A}}{x+a}+\frac{\mathrm{B}}{x+b}+\frac{\mathrm{C}}{x+c}\)

or

P(x) = A(x + b)(x + c) + B(x + a)(x + e) + C(x+a)(x+ b)

It is an identity which is true for all values of x ∈ R.

To find A,put x =-a.

To find B, put x = – b.

To find C, put x = – c.

Thus, the fractions so obtained on the R.H.S. are the partial fractions. e.g.

Let us find the partial fraction of \(\frac{1}{(x-1)(x-2)}\)

∴ \(\frac{1}{(x-1)(x-2)}=\frac{A}{x-1}+\frac{B}{x-2}\)

1 = A(x – 2) + B(x – 1)

Put x = 1, 1 = A(1 – 2) = – A

∴ A = -1

Put x = 2, 1 = B(2 – 1) = B

∴ B = 1

2. Let the linear factor(s) be repeated. Q(x) = (x + a)(x + b)2.

Then,

\(\frac{\mathrm{P}(x)}{\mathrm{Q}(x)}=\frac{\mathrm{A}}{x+a}+\frac{\mathrm{B}}{x+b}+\frac{\mathrm{C}}{(x+b)^{2}}\)

Also, P(x) = A(x + b)2 + B(x + n)(x + b) + C(x + a)

3. Suppose one of the factors of Q(x) be quadratic.

Let Q(x) = (x + a)(x^{2} + bx + c). Then,

\(\frac{\mathrm{P}(x)}{\mathrm{Q}(x)}=\frac{\mathrm{A}}{x+a}+\frac{\mathrm{B} x+\mathrm{C}}{x^{2}+b x+c}\)

Also, P(x) A(x^{2} + b + c) ÷ (x + a)(Bx + c)

Put x = – a, value of A is obtained.

Put x =0, 1, -1, etc., and obtain equations involving A, B, C.

Substitute the value of A in these equations. Then, solve them to find the values of B and C.

OR

Compare the co-efficients of x^{2}, x and constant. Solve the equations so obtained e.g.: Let us find the partial fractions of

\(\frac{1}{(x-1)\left(x^{2}+1\right)}\)

\(\frac{1}{(x-1)\left(x^{2}+1\right)}=\frac{A}{x-1}+\frac{B x+C}{x^{2}+1}\)

∴ 1 = A(x^{2} + 1) + (x – 1)(Bx + C)

1 = A(x^{2} + 1) + B(x^{2} – x) + C(x – 1)

Put x=1, I =A(1 + 1) = 2A

∴ A = \(\frac{1}{2}\).

Comparing the coefficients of x^{2}, we get

O = A + B

∴ B = -A = – \(\frac{1}{2}\)

Comparing the coefficients of x, we get

O = -B + C

∴ C = B = –\(\frac{1}{2}\).

∴ \(\frac{1}{(x-1)\left(x^{2}+1\right)}=\frac{1}{2(x-1)}-\frac{x+1}{2\left(x^{2}+1\right)}\).

Note: It is obvious that by converting the rational function into partial fractions, we can easily integrate the given rational function.

**Integration by Parts**

Let u and v be the functions of x, then

∫uv dx = u∫v dx – ∫u'[∫vdx]dx

= u × Integral of v – Integral of (derivative of u × Integral of v]

Note: Out of two functions, which function is to be considered as first. Usually, we proceed as follows:

∫x^{n} f(x) dx x^{n} = u is First function.

∫(Inverse trig, function) × f(x) dx

Inverse trig. function = u is First function.

∫(log x)f(x) dx, Take log x = u as first function.

**Some Integrals:**

**Definite Integral as the Limit of Sum**

Definite integral as a limit of a sum is defined as

where h = \(\frac{b-a}{n}\). As h → 0, n → ∞.

Note: Some useful series to find the definite integral as the limit of the sum.

**Area Function:**

∫_{a}^{b}f(x) dx is defined as the area of the region bounded by the curve y = f(x), a ≤ x ≤ b,the x-axis and the ordinates x = a and x = b. Let x be a given point in [a, b]. Then,

∫_{a}^{b} f(x) dx represents the area of the shaded region. It is as shmmed that f(x) > 0 for x ∈ [a, b].

This function is denoted by A(x) which is known as area function. i.e.,

A(x) = ∫_{a}^{x} f(x) dx.

→ First Fundamental Theorem of Integral Calculus

Let f be a continuous function on the closed interval [a, b] and let A(x) be the area function. Then, A'(x) =f(x) for all x E [a, b].

→ Second Fundamental Theorem of Integral Calculus

Let f be a continuous function defined on closed interval [a, b] and F be antiderivative of f Then,

∫a b f(x) = [F(x)]ba = F(b) — F(a).

→ Evaluation of Definite Integral by Substitution

- To evaluate ∫a b f(x) dx, let the substitution be x = g(t) such that dx = g'(t) dt.
- Let t = t
_{1}at x = a, t = t_{2}at x = b.

The new limits are t1 to t.

∴ ∫ab f(x) dx = ∫_{t1} ^{t2} f[g(t)g'(t) dt

= F(t_{2}) – F(t_{1}), where

∫f[g(t)]g'(t) dt = F(t).

**Properties of Definite Integrals:**

1. DEFINITION

If \(\frac{d}{d x}\) f(x) = F(x), then ∫F(x) dx = f(x) + c, where ‘c’ is a constant of integration.

2. STANDARD RESULTS

(i). Power Rule. ∫ x^{n} dx = \(\frac{x^{n+1}}{n+1}\) + c, provided n ≠ – 1

(ii) ∫ \(\frac{1}{x}\) dx = log |x| + c, x ≠ 0

(iii) ∫a^{x}dx = \(\frac{a^{x}}{\log a}\) + c, a > 0, a ≠ 1 for all x ∈ R

(iv) ∫ e^{x} dx = e^{x} + c ∀ x ∈ R

(v) ∫ sin x dx= -cosx+c ∀ x ∈ R

(vi) ∫ cos x dx = sin x + c ∀ x ∈ R

(vii) ∫ sec^{2} xdx = tan x + c, x ≠ an odd multiple of \(\frac{\pi}{2}\)

(viii) ∫ cosec^{2} x = – cot x + c, x ≠ an even multiple of \(\frac{\pi}{2}\)

(ix) ∫ sec x tan x dx = sec x + c, x ≠ an odd multiple of \(\frac{\pi}{2}\)

(x) ∫ cosec x cot x dx = – cosec x + c,x ≠ an even multiple of \(\frac{\pi}{2}\)

(xi) ∫ tan x dx – – log | cos x | + c = log |sec x| + c, x ≠ x an odd multiple of \(\frac{\pi}{2}\)

(xii) ∫ cot x dx = log |sin x| + c, x ≠ an even multiple of \(\frac{\pi}{2}\)

(xiii) ∫ sec x dx = log |sec x + tan x| + c, x ≠ an odd multiple of \(\frac{\pi}{2}\)

(xiv) ∫ cosec x dx = log |cosec x – cot x| + c, x ≠ an even multiple of \(\frac{\pi}{2}\).

3. FUNDAMENTAL THEOREMS

(i) ∫ a f(x) dx = a ∫ f(x)dx, where ‘a ’ is any real constant

(ii) ∫ [f_{1}(x) ± f_{2}(x)]dx = ∫ f_{1}(x)dx ± ∫ f_{2}(x)dx

(iii) \(\frac{d}{d x}\)[∫f(x)dx] = f(x)

(iv) ∫\(\frac{f^{\prime}(x)}{f(x)}\)dx = log |f(x)| + c

(v) ∫ f(x))^{n} f'(x)dx = (\(\frac{(f(x))^{n+1}}{n+1}\) + c, n ≠ -1

(vi) ∫ \(\frac{1}{\sqrt{a^{2}-x^{2}}}\)dx = sin^{-1}\(\frac{x}{a}\) + c

(vii) ∫ \(\frac{1}{\sqrt{a^{2}-x^{2}}}\)dx = \(\frac{1}{a} \tan ^{-1} \frac{x}{a}\) + c

(viii) ∫ \(\frac{1}{\sqrt{a^{2}+x^{2}}}\)dx = log|x + \(\sqrt{a^{2}+x^{2}}\)|+ c

(ix) ∫ \(\frac{d x}{\sqrt{x^{2}-a^{2}}}\) = log|x + \(\sqrt{x^{2}-a^{2}}\) |+ c

(x) ∫ \(\sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+c\)

(xi) ∫ \(\sqrt{a^{2}+x^{2}} d x=\frac{x}{2} \sqrt{a^{2}+x^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{a^{2}+x^{2}}\right|+c\)

(xii) ∫ \(\sqrt{x^{2}-a^{2}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}-a^{2}}\right|+c\)

(xiii) ∫ \(\frac{1}{a^{2}-x^{2}}\)dx = \(\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|\)+ c

(xiv) ∫ \(\frac{1}{x^{2}-a^{2}}\)dx = \(\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|\)+ c

4. IMPORTANT RULES

(i) Rule to integrate ∫ sin^{m} x cos^{n} x dx.

(a) If the index of sin x is a positive odd integer, put cos x = t.

(b) If the index of cos x is a positive odd integer, put sin x = t.

(ii) Rule to integrate : \(\int \frac{1}{a \sin ^{2} x+b \cos ^{2} x} d x, \int \frac{1}{a+b \cos ^{2} x} d x\) ; etc.

a sin x + b cos x J a + b cos x

(a) Divide the numerator and denominator by cos^{2}x

(b) Replace sec^{2}x, if any, in the denominator by 1 + tan^{2}x

(c) Put tan x = t so that sec^{2}x dx = dt.

(iii) Integration of Parts.

Integral of the product of two functions = First function x Integral of second – Integral [(diff. coeff. of first) x (integral of second)].

(iv) Rule to integrate \(\int \frac{1}{\text { linear } \sqrt{\text { linear }}} d x\) or \(\int \frac{1}{\text { quadratic } \sqrt{\text { linear }}} d x\) Put \(\sqrt{\text { linear }}=t\)

(v) Rule of integrate \(\int \frac{1}{\text { linear } \sqrt{\text { quadratic }}} d x\) Put linear = \(\frac { 1 }{ t }\)

(vi) Rule to integrate \(\int \frac{x d x}{\text { (Pure Quad.) } \sqrt{\text { Pure Quad }}}\). Put \(\sqrt{\text { Pure Quad }}\) = t

(vii) Rule to integrate \(\int \frac{d x}{\text { (Pure Quad.) } \sqrt{\text { Pure Quad }}}\) Put x = \(\frac{1}{t}\) and \(\sqrt{\text { Pure Quad }}\) = u.

5. Integral As The Limit of a sum

\(\int_{a}^{b}\) f(x)dx = \(\lim _{h \rightarrow 0}\) h[f(a)+f(a + h)+f(a + 2h)+… + f(a + \(\overline{n-1}\)h)], where h = \(\frac{b-a}{n}\).

6. FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS

\(\int_{a}^{b}\)f(x) dx = F(b) – F(a), where ∫ f(x) dx = F(x).

7. PROPERTIES OF DEFINITE INTEGRALS

I. \(\int_{a}^{b}\) f(x)dx = \(\int_{a}^{b}\) f(t) dt

II. \(\int_{a}^{b}\) f(x)dx = –\(\int_{a}^{a}\)f(x)dx. Particular case: \(\int_{a}^{a}\)f(x)dx = 0

III. \(\int_{a}^{b}\) f(x)dx = \(\int_{a}^{b}\) f(a + b – x)dx.

IV. \(\int_{a}^{b}\) f(x)dx = \(\int_{a}^{c}\)f(x)dx + f(x)dx = \(\int_{c}^{b}\) f(x)dx, where a < b < c

V. \(\int_{0}^{a}\) f(x)dx = \(\int_{a}^{b}\) f(a – x)dx

VI. \(\int_{-a}^{a}\) f(x)dx = 0 if f(-x) = -f(x)

= 2\(\int_{0}^{a}\) f(x)dx if f(-x) = f(x)

VII. \(\int_{0}^{2a}\) f(x)dx = \(\int_{0}^{a}\) f(x)dx =\(\int_{0}^{a}\) f(2a – x)dx

VIII. \(\int_{0}^{2a}\) f(x)dx = 2\(\int_{0}^{a}\) f(x) dx

= 0

f(2a-x)dx = f(x)

if (2a – x) = f(x)

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