ISC Class 12 Maths: Integrals – Basic Rules with Easy Examples

Why Integrals Matter in Everyday Life

Ever wondered how engineers find the area under a weird curve or how economists calculate total profit over time? That’s what integrals do – they add up tiny pieces to give a whole.

💡 In Simple Words: An integral is like stacking a bunch of very thin Lego bricks to build a shape. Each brick is a tiny slice, and when you put them together, you get the full picture.

What Is an Integral? (First Time Explanation)

An integral is a mathematical tool that measures the total accumulation of something – area, distance, or even probability. Think of water flowing through a pipe: if you know the flow rate at each instant, adding up (integrating) those rates tells you how much water passed in total.

Basic Integration Rules You Must Know

These rules are the shortcuts you’ll use in almost every exam problem. Memorise the pattern, not the whole proof.

1. Power Rule

If you have xⁿ (x raised to the power n), the integral is (xⁿ⁺¹)/(n+1) + C , where C is the constant of integration (the “+ something” you can add because derivatives erase constants). It works for any n ≠ –1.

2. Constant Multiple Rule

Pull a constant out front. ∫ k·f(x) dx = k·∫ f(x) dx. Like factoring out a common ingredient when cooking.

3. Sum/Difference Rule

Integrate term by term. ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx. It’s the same as adding or subtracting separate piles of sand.

4. Exponential Rule

For the natural exponential eˣ, the integral is itself: ∫ eˣ dx = eˣ + C. The function never changes – like a treadmill that keeps the same speed.

5. Basic Trigonometric Rules

• ∫ sin x dx = –cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec² x dx = tan x + C

These come from how the slopes (derivatives) of sine, cosine, and tangent behave.

Worked Example: Using the Power Rule

Find ∫ (3x⁴ – 5x + 2) dx.

1. Break it into separate integrals: ∫ 3x⁴ dx – ∫ 5x dx + ∫ 2 dx.
2. Apply the constant multiple rule: 3∫ x⁴ dx – 5∫ x dx + 2∫ 1 dx.
3. Use the power rule: 3·(x⁵/5) – 5·(x²/2) + 2·x.
4. Add the constant C: (3/5)x⁵ – (5/2)x² + 2x + C.

Check by differentiating – you should get the original expression back.

Another Example: Trigonometric Integral

Compute ∫ (2 sin x + 4 cos x) dx.

1. Separate the terms: 2∫ sin x dx + 4∫ cos x dx.
2. Integrate: 2(–cos x) + 4(sin x).
3. Write the final answer: –2 cos x + 4 sin x + C.

Quick Reference Table

Common Mistakes to Avoid

• Forgetting the “+ C”. It’s easy to lose the constant when you jump straight to a definite integral.
• Applying the power rule when n = –1. That case gives a logarithm: ∫ x⁻¹ dx = ln|x| + C.
• Mixing up signs in trigonometric integrals – remember ∫ sin x = –cos x, not +.

📝 Likely Exam Questions

1. Find the indefinite integral: ∫ (4x³ – 7) dx.
   Answer: (4/4)x⁴ – 7x + C = x⁴ – 7x + C.
2. Evaluate: ∫ (5 sin θ) dθ.
   Answer: –5 cos θ + C.
3. Integrate using the power rule: ∫ (2x⁻²) dx.
   Answer: 2·(x⁻¹/–1) + C = –2/x + C.
4. Compute the integral: ∫ (eˣ + 3 cos x) dx.
   Answer: eˣ + 3 sin x + C.
5. State the constant of integration and explain why it appears.
   Answer: The constant C represents any fixed number that disappears when differentiating; it ensures the antiderivative is complete.