Seshu cheera

 Here’s the comprehensive guide for Chapter 7: Integrals from Class 12 CBSE Maths: 🧠 A. Key Concepts & Properties Indefinite Integrals ∫ f ( x )   d x = F ( x ) + C \int f(x)\,dx = F(x) + C Basic rules: ∫ x n d x = x n + 1 n + 1 + C \int x^n dx = \frac{x^{n+1}}{n+1} + C ∫ e x d x = e x + C \int e^x dx = e^x + C , ∫ sin ⁡ x   d x = − cos ⁡ x + C \int \sin x\,dx = -\cos x + C , etc. Integration Techniques Substitution : Let u = g ( x ) u = g(x) , then ∫ f ′ ( g ( x ) ) g ′ ( x ) d x = ∫ f ′ ( u ) d u \int f'(g(x))g'(x)dx = \int f'(u)du . Integration by Parts : ∫ u   d v = u v − ∫ v   d u \int u\,dv = uv - \int v\,du Ideal for polynomial×log/trig. Definite Integrals & Fundamental Theorem ∫ a b f ( x )   d x = F ( b ) − F ( a ) \int_a^b f(x)\,dx = F(b) - F(a) Properties: ∫ a a = 0 \int_a^a = 0 , ∫ a b = − ∫ b a \int_a^b = -\int_b^a ∫ a b [ f + g ] = ∫ a b f + ∫ a b g \int_a^b [f+g] = \int_a^b f + \int_a^b g ∫ a b f ( x ) ...

  Class 10 CBSE Maths – Chapter 3: Pair of Linear Equations in Two Variables 🔹 Key Concepts and Formulas ✅ Linear Equation in Two Variables: An equation of the form: a x + b y + c = 0 ax + by + c = 0 Where a, b, c are real numbers, and a and b are not both zero. ✅ Solution of a Pair of Linear Equations: A solution is a pair (x, y) that satisfies both equations simultaneously. ✅ Graphical Method: Each equation represents a straight line. The point of intersection gives the solution. Types of Solutions: Consistent and Unique (Lines intersect at one point) Consistent and Infinite Solutions (Lines coincide) Inconsistent (Lines are parallel) Condition Table: Condition Type of Lines Number of Solutions a 1 a 2 ≠ b 1 b 2 \frac{a_1}{a_2} \neq \frac{b_1}{b_2} Intersecting lines Unique solution a 1 a 2 = b 1 b 2 = c 1 c 2 \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} Coincident lines Infinitely many a 1 a 2 = b 1 b 2 ≠ c 1 c 2 \fra...