📌 1. Introduction to Integrals

Integration is the reverse process of differentiation.

👉 If:$$\frac{d}{dx}(F(x)) = f(x)$$dxd​(F(x))=f(x)

Then:$$\int f(x)\,dx = F(x) + C$$∫f(x)dx=F(x)+C

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$\int f(x)\,dx = F(x) + C$∫f(x)dx=F(x)+C

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📌 Key Idea:

Integration helps us recover the original function from its derivative.

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📖 2. Types of Integrals

🔹 1. Indefinite Integral

$$\int f(x)\,dx = F(x) + C$$∫f(x)dx=F(x)+C

• No limits
• Contains constant of integration

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🔹 2. Definite Integral

$$\int_a^b f(x)\,dx = F(b) – F(a)$$∫ab​f(x)dx=F(b)−F(a)

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$\int_a^b f(x)\,dx = F(b) – F(a)$∫ab​f(x)dx=F(b)−F(a)

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• Has limits
• No constant

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📊 3. Standard Integrals

🔹 Power Rule

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$∫xndx=n+1xn+1​+C(n=−1)

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$\int x^n dx = \frac{x^{n+1}}{n+1} + C$∫xndx=n+1xn+1​+C

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🔹 Logarithmic

$$\int \frac{1}{x} dx = \ln |x| + C$$∫x1​dx=ln∣x∣+C

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🔹 Exponential

$$\int e^x dx = e^x + C$$∫exdx=ex+C

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🔹 Trigonometric

• $\int \sin x dx = -\cos x + C$∫sinxdx=−cosx+C
• $\int \cos x dx = \sin x + C$∫cosxdx=sinx+C
• $\int \sec^2 x dx = \tan x + C$∫sec2xdx=tanx+C

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📌 4. Properties of Integrals

🔹 Linearity

$$\int (af(x) + bg(x)) dx = a\int f(x) dx + b\int g(x) dx$$∫(af(x)+bg(x))dx=a∫f(x)dx+b∫g(x)dx

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$\int (af(x) + bg(x)) dx = a\int f(x) dx + b\int g(x) dx$∫(af(x)+bg(x))dx=a∫f(x)dx+b∫g(x)dx

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🔹 Additivity (Definite Integrals)

$$\int_a^b f(x)dx + \int_b^c f(x)dx = \int_a^c f(x)dx$$∫ab​f(x)dx+∫bc​f(x)dx=∫ac​f(x)dx

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🔁 5. Methods of Integration

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🔹 1. Substitution Method

If:$$\int f(g(x))g'(x) dx$$∫f(g(x))g′(x)dx

Let $t = g(x)$t=g(x)

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🔹 2. Integration by Parts

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$$\int u\,dv = uv – \int v\,du$$∫udv=uv−∫vdu

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$\int u\,dv = uv – \int v\,du$∫udv=uv−∫vdu

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🔹 3. Integration by Partial Fractions

Used for rational functions

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🔹 4. Integration of Trigonometric Functions

Using identities

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📊 6. Definite Integrals Properties

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🔹 1. Limits Same

$$\int_a^a f(x)dx = 0$$∫aa​f(x)dx=0

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🔹 2. Reversing Limits

$$\int_a^b f(x)dx = -\int_b^a f(x)dx$$∫ab​f(x)dx=−∫ba​f(x)dx

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🔹 3. Symmetry

$$\int_{-a}^{a} f(x)dx = \begin{cases} 2\int_0^a f(x)dx & \text{(even)} \\ 0 & \text{(odd)} \end{cases}$$∫−aa​f(x)dx={2∫0a​f(x)dx0​(even)(odd)​

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$\int_{-a}^{a} f(x)dx = 2\int_0^a f(x)dx$∫−aa​f(x)dx=2∫0a​f(x)dx

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📐 7. Fundamental Theorem of Calculus

It connects differentiation and integration.$$\frac{d}{dx} \int_a^x f(t)dt = f(x)$$dxd​∫ax​f(t)dt=f(x)

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$\frac{d}{dx} \int_a^x f(t)dt = f(x)$dxd​∫ax​f(t)dt=f(x)

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📌 8. Applications of Integrals

• Area under curve
• Physics (distance, work)
• Economics
• Probability

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❗ Common Mistakes

• Forgetting constant $C$C
• Wrong substitution
• Incorrect limits
• Sign errors

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🧠 Exam Tips

• Learn standard formulas
• Practice integration methods
• Solve NCERT thoroughly
• Focus on properties

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📚 Practice Questions

1. Evaluate integrals
2. Use substitution
3. Apply integration by parts
4. Solve definite integrals

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🎯 Conclusion

Integrals are a powerful mathematical tool used in various real-life applications. Mastering integration techniques and properties is essential for success in board exams and competitive exams.