Integration Formulas: Comprehensive Formula Sheet for Students

1. Basic Integration Formulas

2. Trigonometric Integration Formulas

3. Integration by Parts Formula (UV Formula)

Formula: ∫u·v dx = u∫v dx – ∫[du/dx · ∫v dx]dx

Alternative Form: ∫u dv = uv – ∫v du

ILATE Rule for Choosing u and v

When applying integration by parts, choose u according to the ILATE priority:

Example Application: ∫x·e^x dx

Here, u = x (Algebraic), dv = e^x dx (Exponential)

Solution: x·e^x – ∫e^x dx = x·e^x – e^x + C = e^x(x-1) + C

4. Special Integration Formulas

5. Exponential and Logarithmic Integration Formulas

6. Inverse Trigonometric Integration Formulas

7. Reduction Formulas (Advanced)

8. Definite Integration Formulas

9. Cauchy Integral Formula (Advanced Calculus)

For Complex Analysis:

If f(z) is analytic inside and on a simple closed contour C, and z₀ is any point inside C, then:

f(z₀) = (1/2πi)∮[C] f(z)/(z-z₀) dz

nth Derivative Formula:

f^(n)(z₀) = (n!/2πi)∮[C] f(z)/(z-z₀)^(n+1) dz

This formula is fundamental in complex analysis and has applications in physics and engineering.

Relationship Between Differentiation and Integration

FAQs about Integration Formula

Q. What is the basic formula of integration?

The most basic formula of integration is:

∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1

This is called the power rule for integration. Here, C represents the constant of integration, which is added because the derivative of any constant is zero. For the special case when n = -1, we have ∫(1/x)dx = ln|x| + C.

Q. What are the formulas for integration?

The essential integration formulas include:

1. Power Rule: ∫x^n dx = x^(n+1)/(n+1) + C
2. Exponential: ∫e^x dx = e^x + C
3. Logarithmic: ∫(1/x)dx = ln|x| + C
4. Trigonometric: ∫sin x dx = -cos x + C; ∫cos x dx = sin x + C
5. Integration by Parts: ∫u·v dx = u∫v dx – ∫[du/dx·∫v dx]dx

These formulas form the foundation for solving most integration problems in calculus.

Q. What is the basic equation of integration?

The basic equation of integration is expressed as:

∫f(x)dx = F(x) + C

Where:

• f(x) is the integrand (function to be integrated)
• F(x) is the antiderivative of f(x)
• C is the constant of integration
• dx indicates integration with respect to x

This equation states that integration finds a function F(x) whose derivative equals f(x).

Q. एकीकरण का मूल सूत्र क्या है?

एकीकरण का सबसे मूल सूत्र है:

∫x^n dx = (x^(n+1))/(n+1) + C, जहाँ n ≠ -1

यह घात नियम (Power Rule) कहलाता है। यहाँ C समाकलन स्थिरांक (constant of integration) है जो इसलिए जोड़ा जाता है क्योंकि किसी भी स्थिरांक का अवकलज शून्य होता है।

अन्य महत्वपूर्ण सूत्र:

• ∫e^x dx = e^x + C
• ∫(1/x)dx = ln|x| + C
• ∫sin x dx = -cos x + C
• ∫cos x dx = sin x + C

Q. कक्षा 12 के भागों द्वारा एकीकरण का सूत्र क्या है?

भागों द्वारा समाकलन (Integration by Parts) का सूत्र है:

∫u·v dx = u∫v dx – ∫[du/dx · ∫v dx]dx

या

∫u dv = uv – ∫v du

ILATE नियम का प्रयोग: u चुनने के लिए ILATE क्रम का पालन करें:

• I = प्रतिलोम त्रिकोणमितीय फलन (Inverse trigonometric)
• L = लघुगणकीय फलन (Logarithmic)
• A = बीजगणितीय फलन (Algebraic)
• T = त्रिकोणमितीय फलन (Trigonometric)
• E = घातांकीय फलन (Exponential)

उदाहरण: ∫x·e^x dx = x·e^x – ∫e^x dx = x·e^x – e^x + C = e^x(x-1) + C

Tips for Mastering Integration Formulas

1. Practice Regularly: Work through problems daily to reinforce formula memorization
2. Understand Patterns: Recognize when to apply specific formulas based on function types
3. Use ILATE Rule: Apply this systematic approach for integration by parts
4. Verify Results: Differentiate your answer to check if you get the original function
5. Create Formula Cards: Make flashcards for quick reference and review
6. Group Similar Formulas: Study related formulas together (all trig formulas, all exponential formulas)
7. Solve Previous Years’ Papers: Practice with actual exam questions for Class 12 boards and competitive exams

Common Mistakes to Avoid

1. Forgetting the constant C in indefinite integrals
2. Misapplying the power rule when n = -1
3. Incorrect sign changes in trigonometric integrations
4. Not simplifying before integrating
5. Wrong choice of u and v in integration by parts
6. Ignoring absolute value in logarithmic results

Conclusion

Regular practice with these formulas, combined with understanding their derivations and applications, will build strong problem-solving skills in calculus. Remember that integration is not just about memorizing formulas but understanding when and how to apply them effectively.