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 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 ) ...

 Here's the complete guide to Chapter 6: Applications of Derivatives (Class 12 CBSE Maths, 2025–26 syllabus): 🧠 A. Key Concepts & Applications Rate of Change The derivative d y / d x dy/dx represents the rate at which y y changes w.r.t. x x . Applicable in physics (velocity, acceleration), economics (marginal cost/revenue), biology, etc. Tangents and Normals Slope of tangent at x = a x = a : m = f ′ ( a ) m = f'(a) . Equation of tangent: y − f ( a ) = f ′ ( a ) ( x − a ) y - f(a) = f'(a)(x - a) . Normal line slope: − 1 / f ′ ( a ) -1/f'(a) . Increasing/Decreasing Functions & Stationary Points If f ′ ( x ) > 0 f'(x) > 0 → increasing; f ′ ( x ) < 0 f'(x) < 0 → decreasing. Stationary points where f ′ ( x ) = 0 f'(x) = 0 . Test using sign of f ′ ( x ) f'(x) or second derivative f ′ ′ ( x ) f''(x) (if > 0 >0 : min, if < 0 <0 : max). Maxima and Minima Locally highes...