📘 Integrals – NCERT Class 12 (Detailed Summary)

🔹 1. Introduction

The chapter Integrals is the reverse process of differentiation 🔄. If differentiation finds the rate of change, integration helps us find the original function. Integrals play a major role in mathematics, physics, economics, and engineering 🚀.

🔹 2. Integration as the Inverse Process of Differentiation

If dy/dx = f(x), then ∫ f(x) dx = y + C. Here, C is the constant of integration ✨ because differentiation removes constants.

Example:
∫ 2x dx = x² + C

🔹 3. Basic Integrals

Some standard integrals you must remember 🧠:

• ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, (n ≠ -1)
• ∫ 1/x dx = ln|x| + C
• ∫ eˣ dx = eˣ + C
• ∫ aˣ dx = aˣ / ln a + C
• ∫ sin x dx = -cos x + C
• ∫ cos x dx = sin x + C

🔹 4. Methods of Integration

📌 (a) Integration by Substitution

This method is used when the integral becomes simpler after changing variables 🔁.

Example:
∫ 2x cos(x²) dx
Put x² = t → dt = 2x dx
Result: ∫ cos t dt = sin t + C

📌 (b) Integration Using Identities

Trigonometric identities are used to simplify integrals 🧩.

Example:
∫ sin²x dx
Use sin²x = (1 - cos2x)/2

📌 (c) Integration by Parts

Used when integrals are products of two functions ✖️.

Formula:
∫ u dv = u v − ∫ v du

Example:
∫ x eˣ dx

📌 (d) Integration Using Partial Fractions

This method is applied when the integrand is a rational function 🧮.

Example:
∫ 1/(x² − 1) dx

🔹 5. Integrals of Some Particular Functions

NCERT includes special forms like:

• ∫ 1/(x² + a²) dx = (1/a) tan⁻¹(x/a) + C
• ∫ 1/√(a² − x²) dx = sin⁻¹(x/a) + C
• ∫ 1/(x² − a²) dx = (1/2a) ln|(x − a)/(x + a)| + C

🔹 6. Definite and Indefinite Integrals (Brief Idea)

This chapter mainly focuses on indefinite integrals, which represent families of curves 🌈. Definite integrals are discussed in the next chapter.

🔹 7. Important Points to Remember

• Always add the constant of integration ➕C
• Choose the correct method based on the integrand 🎯
• Simplify before integrating whenever possible ✨
• Practice standard integrals regularly 📚

🌟 Conclusion

The chapter Integrals builds a strong foundation for applications like area under curves, physics problems, and higher mathematics 🔥. Mastering techniques and formulas is the key to scoring well in exams 💯.