FUZY MATH ACADEMY

Class 12 Maths Chapter 7 Integrals – Exercise 7.3 NCERT Solutions Introduction Exercise 7.3 focuses on integration using partial fractions. Students learn how to decompose rational functions into simpler fractions and integrate them easily. This technique is vital for solving complex integrals, especially in algebraic and calculus problems. Formulas Used Partial Fraction Decomposition: For rational function P ( x ) Q ( x ) , where degree of P ( x ) < degree of Q ( x ) , P ( x ) Q ( x ) = A ( x − a ) + B ( x − b ) + … Integration of Simple Fractions: ∫ 1 x − a d x = ln ⁡ ∣ x − a ∣ + C ∫ 1 ( x − a ) n d x = ( x − a ) 1 − n 1 − n + C , n ≠ 1 Students Frequently Make Mistakes Forgetting to check degree condition before decomposition. Errors in solving coefficients A , B , C . Skipping modulus in logarithmic integrals. Confusing repeated linear factors with distinct ones. Forgetting constant of integration + C . NCERT Questions with Step‑by‑Step Solutions (10) Q1. Evaluate ∫ 1 x 2 −...

Class 12 Maths Chapter 7 Integrals – Exercise 7.2 NCERT Solutions Introduction Exercise 7.2 focuses on integration using standard results and substitution method. Students practice solving integrals of algebraic, trigonometric, exponential, and logarithmic functions by applying substitution and known formulas. This exercise is crucial for building problem‑solving skills in calculus. Formulas Used Substitution Rule: If x = g ( t ) , then ∫ f ( x )   d x = ∫ f ( g ( t ) )   g ′ ( t )   d t Standard Integrals: ∫ 1 1 + x 2 d x = tan ⁡ − 1 x + C ∫ 1 1 − x 2 d x = sin ⁡ − 1 x + C ∫ 1 x x 2 − 1 d x = sec ⁡ − 1 x + C Exponential and Logarithmic: ∫ e x d x = e x + C , ∫ 1 x d x = ln ⁡ ∣ x ∣ + C Students Frequently Make Mistakes Forgetting to change limits in definite integrals after substitution. Skipping differential term d x = g ′ ( t ) d t . Confusing inverse trigonometric integrals. Errors in simplification after substitution. Forgetting constant of integration + C . NCERT Questions with...

Class 12 Maths Chapter 7 Integrals – Exercise 7.1 NCERT Solutions Introduction Exercise 7.1 introduces the concept of indefinite integrals as the reverse process of differentiation. Students learn basic integration formulas, rules of integration, and how to apply them to algebraic and trigonometric functions. This exercise builds the foundation for definite integrals and applications. Formulas Used Basic Integration Formulas: ∫ x n d x = x n + 1 n + 1 + C , n ≠ − 1 ∫ 1 x d x = ln ⁡ ∣ x ∣ + C ∫ e x d x = e x + C ∫ sin ⁡ x d x = − cos ⁡ x + C ∫ cos ⁡ x d x = sin ⁡ x + C ∫ sec ⁡ 2 x d x = tan ⁡ x + C ∫ csc ⁡ 2 x d x = − cot ⁡ x + C ∫ sec ⁡ x tan ⁡ x d x = sec ⁡ x + C ∫ csc ⁡ x cot ⁡ x d x = − csc ⁡ x + C Constant Rule: ∫ k f ( x ) d x = k ∫ f ( x ) d x Sum Rule: ∫ [ f ( x ) + g ( x ) ] d x = ∫ f ( x ) d x + ∫ g ( x ) d x Students Frequently Make Mistakes Forgetting the constant of integration + C . Confusing integration with differentiation. Applying power rule incorrectly when n = − 1 . ...

Class 12 Maths Chapter 8 Application of Integrals – Exercise 8.3 NCERT Solutions Introduction Exercise 8.3 focuses on finding areas bounded by curves and lines using definite integrals in more complex cases. Students learn how to calculate areas enclosed between curves, coordinate axes, and intersecting lines. This exercise is essential for mastering applications of integrals in geometry and real‑life contexts. Formulas Used Area between two curves y = f ( x ) and y = g ( x ) : A = ∫ a b [ f ( x ) − g ( x ) ]   d x , f ( x ) ≥ g ( x ) Area bounded by curve and x‑axis: A = ∫ a b ∣ f ( x ) ∣   d x Area bounded by curve and y‑axis: A = ∫ c d f ( y )   d y Symmetry Property: If curve is symmetric, calculate area for half and double it. Students Frequently Make Mistakes Forgetting to find intersection points of curves correctly. Errors in setting integration limits. Confusing which curve is upper/lower in given interval. Ignoring absolute value when curve lies below x‑axis. Skipping sym...

Class 12 Maths Chapter 8 Application of Integrals – Exercise 8.2 NCERT Solutions Introduction Exercise 8.2 focuses on finding areas bounded by curves and lines using definite integrals. Students learn how to calculate the area enclosed between two curves, the coordinate axes, and given limits. This exercise is essential for geometry, calculus, and real‑life applications such as physics and economics. Formulas Used Area between two curves y = f ( x ) and y = g ( x ) : A = ∫ a b [ f ( x ) − g ( x ) ]   d x , f ( x ) ≥ g ( x ) Area bounded by curve and x‑axis: A = ∫ a b ∣ f ( x ) ∣   d x Area bounded by curve and y‑axis: A = ∫ c d f ( y )   d y Students Frequently Make Mistakes Forgetting to identify which curve is above the other. Errors in setting correct limits of integration. Confusing absolute value when curve lies below x‑axis. Skipping symmetry property to simplify calculations. Misinterpreting geometric meaning of definite integral. NCERT Questions with Step‑by‑Step Solutions (10...