All Important Formula — Class 12 Maths Notes

All Important Formula · Class 12 Maths · 6 topics.

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Topics covered in All Important Formula

1.Algebraic Formulas:
Algebraic Formulas:
1. (a+b)2=a2+2ab+b2
  - Expansion of a binomial square (sum of squares).
2. (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2
  - Expansion of a binomial square (difference of squares).
3. (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2
  - Difference of squares formula.
4. a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)
  - Sum of cubes.
5. a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
  - Difference of cubes.
6. (a−b2)2=a2−2ab+b24\left(\frac{a - b}{2}\right)^2 = \frac{a^2 - 2ab + b^2}{4}
  - Square of a fraction involving binomial subtraction.
7. a3−b3=(a−b)(a2+ab+b2)+3ab(a−b)a^3 - b^3 = (a - b)(a^2 + ab + b^2) + 3ab(a - b)
  - Another variation for the difference of cubes expansion.
8. (a−b)4=a4−4a3b+6a2b2−4ab3+b4(a - b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4
  - Fourth power binomial expansion (difference).
9. a3+b3=(a+b)3−3ab(a+b)a^3 + b^3 = (a + b)^3 - 3ab(a + b)
  - Sum of cubes in terms of a binomial cube.
10. (a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
  - Cube of a binomial (sum).
11. (a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
  - Cube of a binomial (difference).
12. (a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
  - Repeated cube of a binomial (difference).
Explanation and Example of Key Formula

Formula: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
- Components:
  - aa and bb are variables or numbers.
  - The square of the sum of aa and bb expands to a2+2ab+b2a^2 + 2ab + b^2.
Example:
Let's expand (3+2)2(3 + 2)^2:
1. Given: a=3a = 3, b=2
  
  b = 2
2. Plug into the formula: (3+2)2=32+2×3×2+22(3 + 2)^2 = 3^2 + 2 \times 3 \times 2 + 2^2
3. Simplify: =9+12+4=25
2.Factors Formula (Logarithmic Formulas)
1. ax=N ⟹ x=log⁡aNa^x = N \implies x = \log_a N
- Explanation: If aa raised to the power xx equals NN, then xx is the logarithm of NN with base aa.
- Example:
  If 23=82^3 = 8, then log⁡28=3\log_2 8 = 3.
2. log⁡aa=1\log_a a = 1
- Explanation: The logarithm of a number to its own base is always 1.
- Example:
  log⁡55=1\log_5 5 = 1.
3. log⁡1010=1\log_{10} 10 = 1
- Explanation: The logarithm of 10 to the base 10 is always 1.
- Example:
  log⁡1010=1\log_{10} 10 = 1.
4. log⁡a(m×n)=log⁡am+log⁡an\log_a (m \times n) = \log_a m + \log_a n
- Explanation: The logarithm of the product of two numbers is the sum of their logarithms to the same base.
- Example:
  
  Find log⁡2(8×4)\log_2 (8 \times 4): log⁡2(8×4)=log⁡28+log⁡24\log_2 (8 \times 4) = \log_2 8 + \log_2 4 Since log⁡28=3\log_2 8 = 3 and log⁡24=2\log_2 4 = 2: log⁡2(8×4)=3+2=5\log_2 (8 \times 4) = 3 + 2 = 5
5. log⁡a(mn)=log⁡am−log⁡an\log_a \left(\frac{m}{n}\right) = \log_a m - \log_a n
- Explanation: The logarithm of the division of two numbers is the difference of their logarithms.
- Example:
  
  Find log⁡2(84)\log_2 \left(\frac{8}{4}\right): log⁡2(84)=log⁡28−log⁡24\log_2 \left(\frac{8}{4}\right) = \log_2 8 - \log_2 4 Since log⁡28=3\log_2 8 = 3 and log⁡24=2\log_2 4 = 2: log⁡2(84)=3−2=1\log_2 \left(\frac{8}{4}\right) = 3 - 2 = 1
6. log⁡amn=nlog⁡am\log_a m^n = n \log_a m
- Explanation: The logarithm of a power is the exponent times the logarithm of the base number.
- Example:
  Find log⁡282\log_2 8^2: log⁡282=2log⁡28\log_2 8^2 = 2 \log_2 8 Since log⁡28=3\log_2 8 = 3: log⁡282=2×3=6\log_2 8^2 = 2 \times 3 = 6
7. log⁡ax2=2log⁡a∣x∣\log_a x^2 = 2 \log_a |x|
- Explanation: The logarithm of x2x^2 is twice the logarithm of the absolute value of xx.
- Example:
  
  Find log⁡2(−4)2\log_2 (-4)^2: log⁡2(−4)2=2log⁡2∣4∣\log_2 (-4)^2 = 2 \log_2 |4| Since log⁡24=2\log_2 4 = 2: log⁡2(−4)2=2×2=4\log_2 (-4)^2 = 2 \times 2 = 4
8. log⁡ba×log⁡ab=1\log_b a \times \log_a b = 1
- Explanation: The product of log⁡ba\log_b a and log⁡ab\log_a b is always equal to 1.
- Example:
  
  If log⁡39=2\log_3 9 = 2, then log⁡93=12\log_9 3 = \frac{1}{2}, and their product: log⁡39×log⁡93=2×12=1\log_3 9 \times \log_9 3 = 2 \times \frac{1}{2} = 1
9. log⁡ab=1log⁡ba\log_a b = \frac{1}{\log_b a}
- Explanation: The logarithm of bb with base aa is the reciprocal of the logarithm of aa with base bb.
- Example:
  If log⁡28=3\log_2 8 = 3, then log⁡82=13\log_8 2 = \frac{1}{3}.
10. log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}
- Explanation: This is the change of base formula, which allows you to convert logarithms from one base to another.
- Example:
  Find log⁡28\log_2 8 using base 10: log⁡28=log⁡108log⁡102\log_2 8 = \frac{\log_{10} 8}{\log_{10} 2} Approximate values: log⁡108≈0.903,log⁡102≈0.301\log_{10} 8 \approx 0.903, \quad \log_{10} 2 \approx 0.301 So, log⁡28≈0.9030.301=3\log_2 8 \approx \frac{0.903}{0.301} = 3
11. log⁡ba=1log⁡ab\log_b a = \frac{1}{\log_a b}
- Explanation: This is a repetition of Formula 9, stating the reciprocal relationship of logs.
- Example:
  
  If log⁡525=2\log_5 25 = 2, then log⁡255=12\log_{25} 5 = \frac{1}{2}.
3.Trigonometric Ratios and Formulas
📏 Trigonometric Ratios and Formulas
1. Trigonometric Ratios:
  - tan⁡θ=sin⁡θcos⁡θ
    
    \tan \theta = \frac{\sin \theta}{\cos \theta}
  - cot⁡θ=cos⁡θsin⁡θ
    
    \cot \theta = \frac{\cos \theta}{\sin \theta}
  - sec⁡θ=1cos⁡θ
    
    \sec \theta = \frac{1}{\cos \theta}
  - csc⁡θ=1sin⁡θ
    
    \csc \theta = \frac{1}{\sin \theta}
2. Pythagorean Identities:
  - sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  - sec⁡2θ−tan⁡2θ=1\sec^2 \theta - \tan^2 \theta = 1
  - csc⁡2θ−cot⁡2θ=1
    
    \csc^2 \theta - \cot^2 \theta = 1
3. Co-function Identities:
  - sin⁡θ=cos⁡(90∘−θ)\sin \theta = \cos (90^\circ - \theta)
  - cos⁡θ=sin⁡(90∘−θ)\cos \theta = \sin (90^\circ - \theta)
  - tan⁡θ=cot⁡(90∘−θ)\tan \theta = \cot (90^\circ - \theta)
  - cot⁡θ=tan⁡(90∘−θ)\cot \theta = \tan (90^\circ - \theta)
  - sec⁡θ=csc⁡(90∘−θ)\sec \theta = \csc (90^\circ - \theta)
  - csc⁡θ=sec⁡(90∘−θ)
    
    \csc \theta = \sec (90^\circ - \theta)
4. Additional Identities:
  - tan⁡θ×cot⁡θ=1\tan \theta \times \cot \theta = 1
📊 Trigonometric Table for Common Angles

These formulas and table help solve problems related to trigonometry, geometry, and real-world applications like physics, architecture, and engineering.
4.Differentiation Formulas
Basic Differentiation Formulas
1. Power Rule
  ddx(xn)=nxn−1\frac{d}{dx}(x^n) = n x^{n-1}
2. Exponential Function
  ddx(ex)=ex\frac{d}{dx}(e^x) = e^x
3. Logarithmic Differentiation
  ddx(log⁡ax)=1xlog⁡a
  
  \frac{d}{dx}(\log_a x) = \frac{1}{x \log a}
4. Natural Logarithm
  ddx(log⁡ex)=1x
  
  \frac{d}{dx}(\log_e x) = \frac{1}{x}
5. Exponential with Base aa
  ddx(ax)=axlog⁡a\frac{d}{dx}(a^x) = a^x \log a
Trigonometric Functions
1. Derivative of sin⁡x\sin x
  ddx(sin⁡x)=cos⁡x
  
  \frac{d}{dx}(\sin x) = \cos x
2. Derivative of cos⁡x\cos x
  ddx(cos⁡x)=−sin⁡x
  
  \frac{d}{dx}(\cos x) = -\sin x
3. Derivative of tan⁡x\tan x
  ddx(tan⁡x)=sec⁡2x
  
  \frac{d}{dx}(\tan x) = \sec^2 x
4. Derivative of cot⁡x\cot x
  ddx(cot⁡x)=−csc⁡2x
  
  \frac{d}{dx}(\cot x) = -\csc^2 x
5. Derivative of sec⁡x\sec x
  ddx(sec⁡x)=sec⁡xtan⁡x
  
  \frac{d}{dx}(\sec x) = \sec x \tan x
6. Derivative of csc⁡x\csc x
  ddx(csc⁡x)=−csc⁡xcot⁡x\frac{d}{dx}(\csc x) = -\csc x \cot x
Inverse Trigonometric Functions
1. Derivative of sin⁡−1x\sin^{-1} x
  ddx(sin⁡−1x)=11−x2,for −1<x<1
  
  \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}}, \quad \text{for } -1 < x < 1
2. Derivative of cos⁡−1x\cos^{-1} x
  ddx(cos⁡−1x)=−11−x2,for −1<x<1
  
  \frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}}, \quad \text{for } -1 < x < 1
3. Derivative of tan⁡−1x\tan^{-1} x
  ddx(tan⁡−1x)=11+x2\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2}
4. Derivative of cot⁡−1x\cot^{-1} x
  ddx(cot⁡−1x)=−11+x2\frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1 + x^2}
5. Derivative of sec⁡−1x\sec^{-1} x
  ddx(sec⁡−1x)=1∣x∣x2−1,∣x∣>1
  
  \frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x| \sqrt{x^2 - 1}}, \quad |x| > 1
6. Derivative of csc⁡−1x\csc^{-1} x
  ddx(csc⁡−1x)=−1∣x∣x2−1,∣x∣>1\frac{d}{dx}(\csc^{-1} x) = -\frac{1}{|x| \sqrt{x^2 - 1}}, \quad |x| > 1
Hyperbolic Trigonometric Functions
1. Derivative of sinh⁡x\sinh x
  ddx(sinh⁡x)=cosh⁡x
  
  \frac{d}{dx}(\sinh x) = \cosh x
2. Derivative of cosh⁡x\cosh x
  ddx(cosh⁡x)=sinh⁡x
  
  \frac{d}{dx}(\cosh x) = \sinh x
3. Derivative of tanh⁡x\tanh x
  ddx(tanh⁡x)=sech⁡2x
  
  \frac{d}{dx}(\tanh x) = \operatorname{sech}^2 x
4. Derivative of coth⁡x\coth x
  ddx(coth⁡x)=−csch⁡2x
  
  \frac{d}{dx}(\coth x) = -\operatorname{csch}^2 x
5. Derivative of sech⁡x\operatorname{sech} x
  ddx(sech⁡x)=−sech⁡xtanh⁡x
  
  \frac{d}{dx}(\operatorname{sech} x) = -\operatorname{sech} x \tanh x
6. Derivative of csch⁡x\operatorname{csch} x
  ddx(csch⁡x)=−csch⁡xcoth⁡x
5.Integration Formulas
1. Basic Formulas
1. ∫1 dx=x+C
  
  \int 1 \, dx = x + C
2. ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C (for n≠−1n \neq -1)
3. ∫ex dx=ex+C
  
  \int e^x \, dx = e^x + C
4. ∫ax dx=axln⁡a+C\int a^x \, dx = \frac{a^x}{\ln a} + C (for a>0,a≠1a > 0, a \neq 1)
5. ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln |x| + C
2. Trigonometric Formulas
1. ∫sin⁡x dx=−cos⁡x+C
  
  \int \sin x \, dx = -\cos x + C
2. ∫cos⁡x dx=sin⁡x+C
  
  \int \cos x \, dx = \sin x + C
3. ∫sec⁡2x dx=tan⁡x+C
  
  \int \sec^2 x \, dx = \tan x + C
4. ∫csc⁡2x dx=−cot⁡x+C
  
  \int \csc^2 x \, dx = -\cot x + C
5. ∫sec⁡xtan⁡x dx=sec⁡x+C
  
  \int \sec x \tan x \, dx = \sec x + C
6. ∫csc⁡xcot⁡x dx=−csc⁡x+C\int \csc x \cot x \, dx = -\csc x + C
3. Inverse Trigonometric Formulas
1. ∫11−x2 dx=sin⁡−1x+C
  
  \int \frac{1}{\sqrt{1-x^2}} \, dx = \sin^{-1}x + C
2. ∫−11−x2 dx=cos⁡−1x+C
  
  \int \frac{-1}{\sqrt{1-x^2}} \, dx = \cos^{-1}x + C
3. ∫11+x2 dx=tan⁡−1x+C
  
  \int \frac{1}{1+x^2} \, dx = \tan^{-1}x + C
4. ∫−11+x2 dx=cot⁡−1x+C
  
  \int \frac{-1}{1+x^2} \, dx = \cot^{-1}x + C
5. ∫1xx2−1 dx=sec⁡−1∣x∣+C
  
  \int \frac{1}{x\sqrt{x^2-1}} \, dx = \sec^{-1}|x| + C
6. ∫−1xx2−1 dx=csc⁡−1∣x∣+C\int \frac{-1}{x\sqrt{x^2-1}} \, dx = \csc^{-1}|x| + C
4. Exponential and Logarithmic Functions
1. ∫ex dx=ex+C
  
  \int e^x \, dx = e^x + C
2. ∫ln⁡x dx=xln⁡x−x+C
  
  \int \ln x \, dx = x\ln x - x + C
3. ∫ax dx=axln⁡a+C\int a^x \, dx = \frac{a^x}{\ln a} + C
5. Special Forms
1. ∫1a2+x2 dx=1atan⁡−1xa+C
  
  \int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \tan^{-1} \frac{x}{a} + C
2. ∫1a2−x2 dx=sin⁡−1xa+C
  
  \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1} \frac{x}{a} + C
3. ∫1xx2−a2 dx=1asec⁡−1∣x∣a+C\int \frac{1}{x\sqrt{x^2 - a^2}} \, dx = \frac{1}{a} \sec^{-1} \frac{|x|}{a} + C
6. Integration by Substitution
If u=f(x)u = f(x), then:
∫f′(x)g(f(x)) dx=∫g(u) du\int f'(x)g(f(x)) \, dx = \int g(u) \, du
7. Integration by Parts
∫uv dx=u∫v dx−∫(dudx∫v dx)dx\int u v \, dx = u \int v \, dx - \int \left( \frac{du}{dx} \int v \, dx \right) dx
Where uu and vv are functions of xx.
8. Definite Integral Properties
1. ∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a), where F(x)F(x) is the antiderivative of f(x)f(x).
2. ∫aaf(x) dx=0
  
  \int_a^a f(x) \, dx = 0
3. ∫abf(x) dx=−∫baf(x) dx
6.Exponent Laws
Exponent Laws
1. a0=1a^0 = 1
  - Explanation: Any non-zero number raised to the power of 00 is always 11.
2. ap⋅aq=ap+qa^p \cdot a^q = a^{p+q}
  - Explanation: When multiplying two numbers with the same base, add the exponents.
3. (ap)q=ap⋅q(a^p)^q = a^{p \cdot q}
  - Explanation: When raising a power to another power, multiply the exponents.
4. apaq=ap−q\frac{a^p}{a^q} = a^{p-q}
  - Explanation: When dividing two numbers with the same base, subtract the exponents.
5. a−n=1ana^{-n} = \frac{1}{a^n}
  - Explanation: A negative exponent means the reciprocal of the number raised to the positive exponent.
6. a=a1/2\sqrt{a} = a^{1/2}
  - Explanation: The square root of a number can be expressed as the number raised to the power 1/21/2.
7. a3=a1/3\sqrt[3]{a} = a^{1/3}
  - Explanation: The cube root of a number can be expressed as the number raised to the power 1/31/3.
8. an=a1/n\sqrt[n]{a} = a^{1/n}
  - Explanation: The nn-th root of a number can be expressed as the number raised to the power 1/n1/n.
9. a1=aa^1 = a
  - Explanation: Any number raised to the power 11 is the number itself.

All Important Formula — Class 12 Maths Notes

Topics covered in All Important Formula

1.Algebraic Formulas:

Algebraic Formulas:

Explanation and Example of Key Formula

Formula: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

Example:

2.Factors Formula (Logarithmic Formulas)

1. ax=N ⟹ x=log⁡aNa^x = N \implies x = \log_a N

2. log⁡aa=1\log_a a = 1

3. log⁡1010=1\log_{10} 10 = 1

4. log⁡a(m×n)=log⁡am+log⁡an\log_a (m \times n) = \log_a m + \log_a n

5. log⁡a(mn)=log⁡am−log⁡an\log_a \left(\frac{m}{n}\right) = \log_a m - \log_a n

6. log⁡amn=nlog⁡am\log_a m^n = n \log_a m

7. log⁡ax2=2log⁡a∣x∣\log_a x^2 = 2 \log_a |x|

8. log⁡ba×log⁡ab=1\log_b a \times \log_a b = 1

9. log⁡ab=1log⁡ba\log_a b = \frac{1}{\log_b a}

10. log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}

11. log⁡ba=1log⁡ab\log_b a = \frac{1}{\log_a b}

3.Trigonometric Ratios and Formulas

📏 Trigonometric Ratios and Formulas

📊 Trigonometric Table for Common Angles

4.Differentiation Formulas

Basic Differentiation Formulas

Trigonometric Functions

Inverse Trigonometric Functions

Hyperbolic Trigonometric Functions

5.Integration Formulas

2. Trigonometric Formulas

3. Inverse Trigonometric Formulas

4. Exponential and Logarithmic Functions

5. Special Forms

6. Integration by Substitution

7. Integration by Parts

8. Definite Integral Properties

6.Exponent Laws

Exponent Laws

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