⭐ Sabse Important Points · एक नज़र में
- Standard Integrals: Memorize basic forms like ∫x^n dx = x^(n+1)/(n+1) + C (n ≠ -1), ∫1/x dx = ln|x| + C, ∫e^x dx = e^x + C, ∫sin x dx = -cos x + C, etc.
- Fundamental Theorem: ∫_a^b f(x) dx = F(b) - F(a), where F is an antiderivative of f.
- Variable Separable: If dy/dx = f(x)g(y), separate as dy/g(y) = f(x)dx and integrate both sides.
- Integration by Substitution: Use when integrand contains a function and its derivative. Let u = g(x), then du = g'(x)dx.
- Property of Even/Odd Functions: If f is even, ∫_{-a}^{a} f(x) dx = 2∫_0^a f(x) dx; if f is odd, integral is 0.
- Homogeneous Equation: If dy/dx = F(y/x), substitute y = vx, then dy/dx = v + x dv/dx, reducing to separable form.
Introduction · परिचय
Integral Calculus and Differential Equations form a key part of NDA Mathematics. This topic covers indefinite and definite integrals, methods of integration, and solving first-order differential equations. Mastery of standard formulas and techniques is essential for scoring well.
Indefinite Integrals and Standard Methods
Indefinite integrals represent the family of antiderivatives of a function. The fundamental methods include substitution, integration by parts, and partial fractions. Recognizing the form of the integrand is crucial for choosing the right method.
📐 Formula / Rule Box · सूत्र याद रखो
Power Rule
∫x^n dx = x^(n+1)/(n+1) + C, n ≠ -1
Integration by Parts
∫u dv = uv - ∫v du
LIATE Rule
Priority for u: Log, Inverse trig, Algebraic, Trig, Exponential
⭐ Key Points · याद रखने वाली बातें
- Standard Integrals: Memorize basic forms like ∫x^n dx = x^(n+1)/(n+1) + C (n ≠ -1), ∫1/x dx = ln|x| + C, ∫e^x dx = e^x + C, ∫sin x dx = -cos x + C, etc.
- Integration by Substitution: Use when integrand contains a function and its derivative. Let u = g(x), then du = g'(x)dx.
- Integration by Parts: ∫u dv = uv - ∫v du. Choose u by LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).
- Partial Fractions: For rational functions, decompose into simpler fractions before integrating.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Evaluate ∫x e^x dx
✅ Solution: x e^x - e^x + C
💡 Explanation: Use integration by parts: let u = x, dv = e^x dx. Then du = dx, v = e^x. ∫x e^x dx = x e^x - ∫e^x dx = x e^x - e^x + C.
Definite Integrals and Properties
Definite integrals compute the net area under a curve between limits. Key properties simplify evaluation, especially for even/odd functions and periodic functions.
📐 Formula / Rule Box · सूत्र याद रखो
Fundamental Theorem
∫_a^b f(x) dx = F(b) - F(a)
Even Function
∫_{-a}^{a} f(x) dx = 2∫_0^a f(x) dx
Odd Function
∫_{-a}^{a} f(x) dx = 0
Fundamental Theorem
∫_a^b f(x) dx = F(b) - F(a), where F is an antiderivative of f.
⭐ Key Points · याद रखने वाली बातें
- Property of Even/Odd Functions: If f is even, ∫_{-a}^{a} f(x) dx = 2∫_0^a f(x) dx; if f is odd, integral is 0.
- King's Property: ∫_a^b f(x) dx = ∫_a^b f(a+b-x) dx. Useful for trigonometric integrals.
- Area Interpretation: Definite integral gives signed area; for area between curves, use ∫|f(x)-g(x)| dx.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Evaluate ∫_{-π/2}^{π/2} sin^3 x dx
✅ Solution: 0
💡 Explanation: sin^3 x is an odd function because sin(-x) = -sin x, so sin^3(-x) = -sin^3 x. Integral over symmetric limits is 0.
First-Order Differential Equations
First-order differential equations involve derivatives of the first order. The main types are variable separable, homogeneous, and linear equations. Solving them requires identifying the type and applying the standard method.
📐 Formula / Rule Box · सूत्र याद रखो
Linear DE Solution
y·e^{∫P dx} = ∫Q·e^{∫P dx} dx + C
Integrating Factor
IF = e^{∫P dx}
Homogeneous Equation
If dy/dx = F(y/x), substitute y = vx, then dy/dx = v + x dv/dx, reducing to separable form.
Linear Differential Equation
Standard form dy/dx + P(x)y = Q(x). Integrating factor (IF) = e^{∫P dx}. Solution: y·IF = ∫Q·IF dx + C.
Exact Equation
If M dx + N dy = 0 and ∂M/∂y = ∂N/∂x, solution is ∫M dx + ∫(terms of N free from x) dy = C.
⭐ Key Points · याद रखने वाली बातें
- Variable Separable: If dy/dx = f(x)g(y), separate as dy/g(y) = f(x)dx and integrate both sides.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Solve dy/dx = x/y
✅ Solution: y^2 = x^2 + C
💡 Explanation: Separate variables: y dy = x dx. Integrate: y^2/2 = x^2/2 + C1, so y^2 = x^2 + C (where C = 2C1).
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1Evaluate ∫ (2x + 3)^5 dx
👀 Answer dekho
✅ Answer: (2x + 3)^6 / 12 + C
💡 Explanation: Let u = 2x + 3, then du = 2 dx, so dx = du/2. Integral becomes (1/2)∫u^5 du = (1/2)(u^6/6) + C = u^6/12 + C = (2x+3)^6/12 + C.
Q2Find the integrating factor of dy/dx + (2/x)y = x^2.
👀 Answer dekho
✅ Answer: x^2
💡 Explanation: P(x) = 2/x. IF = e^{∫(2/x) dx} = e^{2 ln x} = e^{ln x^2} = x^2.
Q3Evaluate ∫_0^π sin x dx.
👀 Answer dekho
✅ Answer: 2
💡 Explanation: ∫ sin x dx = -cos x. Evaluate from 0 to π: (-cos π) - (-cos 0) = (-(-1)) - (-1) = 1 + 1 = 2.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Forgetting the constant of integration C in indefinite integrals. Always add C.
- Misapplying the LIATE rule for integration by parts; choose u according to LIATE to simplify the integral.
- In definite integrals, ignoring symmetry properties that can simplify the problem; check if the integrand is even or odd.
- In differential equations, not converting to standard linear form before finding the integrating factor; ensure the coefficient of dy/dx is 1.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Standard Integrals: Memorize basic forms like ∫x^n dx, ∫1/x dx, ∫e^x dx, ∫sin x dx, ∫cos x dx, etc.
- Integration by Parts: Formula ∫u dv = uv - ∫v du; choose u by LIATE.
- Definite Integral Properties: Use even/odd symmetry and King's property to simplify integrals.
- Linear Differential Equation: Standard form dy/dx + P(x)y = Q(x); IF = e^{∫P dx}; solution y·IF = ∫Q·IF dx + C.
- Variable Separable: Separate variables and integrate both sides; don't forget constant C.