Integrals Class 12 Exam Prep Revision — Grandmaster Guide
Ayush (Founder)
Exam Strategist
- ∫(1/x) dx = ln|x| + C
- ∫aⁿ da = (aⁿ⁺¹)/(n+1) + C
- ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C
- ∫eˣ dx = eˣ + C
- ∫s∈(x) dx = -cos(x) + C
- ∫cos(x) dx = s∈(x) + C
- ∫tan(x) dx = -ln|cos(x)| + C
- ∫sec²(x) dx = tan(x) + C
- ∫cosec(x)cot(x) dx = -cosec(x) + C
- ∫(1/√(x²-a²)) dx = ln|x + √(x²-a²)| + C
- ∫(1/√(a²-x²)) dx = arcs∈(x/a) + C
- ∫(1/(a²+x²)) dx = (1/a)arctan(x/a) + C
🪤 The 5 Mistakes That Cost Marks
- Not checking the limits of integration
- Forgetting to add the constant of integration
- Not using the correct substitution method
- Incorrectly applying integration by parts
- Not simplifying the integral before evaluating it
✏️ 3 Solved PYQs
- Question 1: Evaluate ∫(x²+1)/(x+1) dx Step 1: Factor the numerator as (x+1)(x-1) + 2 Step 2: Simplify the integral to ∫(x-1) dx + ∫2/(x+1) dx Step 3: Evaluate the integral as (x²/2) - x + 2ln|x+1| + C
- Question 2: Evaluate ∫(x+1)√(x²+2x+1) dx Step 1: Substitute u = x² + 2x + 1, du/dx = 2x + 2 Step 2: Simplify the integral to (1/2)∫√u du Step 3: Evaluate the integral as (1/3)u³/² + C Step 4: Substitute back u = x² + 2x + 1
- Question 3: Evaluate ∫(1/(x²+4x+5)) dx Step 1: Complete the square ∈ the denominator as (x+2)² + 1 Step 2: Substitute u = x + 2, du/dx = 1 Step 3: Simplify the integral to ∫(1/(u²+1)) du Step 4: Evaluate the integral as arctan(u) + C Step 5: Substitute back u = x + 2
🧠 The One Thing Most Students Get Wrong
- Most students struggle with integration by parts, specifically with choosing the correct u and dv
- The key is to choose u as the function that simplifies when differentiated, and dv as the function that simplifies when integrated
- For example, ∈ the integral ∫xs∈(x) dx, choose u = x and dv = s∈(x) dx
👁️ Ayush's Note
- When dealing with definite integrals, make sure to evaluate the integral at the upper and lower limits of integration
- Use the formula ∫ₐᵇ f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x)
- For example, ∈ the integral ∫₀ᵖᵢ s∈(x) dx, evaluate the integral as -cos(x) |₀ᵖᵢ = -cos(π) + cos(0) = 2
🔁 Last 5 Minutes Box
- Check all limits of integration
- Verify all constants of integration
- Review all substitution methods
- Double-check all integration by parts
- Simplify all final answers
📝 Practice MCQs
1. What is the value of ∫(1/x) dx?
A) ln|x|
B) ln|x| + C
C) 1/x
D) x
Answer: B) The integral of 1/x is ln|x| + C, where C is the constant of integration.
2. Evaluate ∫x² dx
A) (x³/3) + C
B) (x²/2) + C
C) x³ + C
D) (x³/2) + C
Answer: A) The integral of x² is (x³/3) + C, using the power rule of integration.
3. What is the value of ∫eˣ dx?
A) eˣ
B) eˣ + C
C) 1/e
D) e⁻ˣ
Answer: B) The integral of eˣ is eˣ + C, where C is the constant of integration.
4. Evaluate ∫(1/√(x²-a²)) dx
A) ln|x + √(x²-a²)|
B) ln|x + √(x²-a²)| + C
C) √(x²-a²)
D) 1/√(x²-a²)
Answer: B) The integral of 1/√(x²-a²) is ln|x + √(x²-a²)| + C, using the substitution method.
5. What is the value of ∫(1/(a²+x²)) dx?
A) (1/a)arctan(x/a)
B) (1/a)arctan(x/a) + C
C) 1/(a²+x²)
D) x/(a²+x²)
Answer: B) The integral of 1/(a²+x²) is (1/a)arctan(x/a) + C, using the substitution method.
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📚 Academic References
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This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.
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