IIT-JEE / UPSC / Railways Case Studies 813+ high-difficulty problems — IIT-JEE (179), UPSC (188), Indian Railways (230) and SSC-CGL (216) — each with a full step-by-step worked solution using the formulas on FormulaWon. Master the techniques that top the toughest competitive exams in India.
Evaluate ∫ 1 3 ( 4 x 2 + 3 x ) d x \int_{1}^{3} (4x^2 + 3x)\,dx ∫ 1 3 ( 4 x 2 + 3 x ) d x . Uses Definite Integral
Evaluate ∫ 1 4 ( 5 x 2 + 5 x ) d x \int_{1}^{4} (5x^2 + 5x)\,dx ∫ 1 4 ( 5 x 2 + 5 x ) d x . Uses Definite Integral
Evaluate ∫ 0 2 ( 5 x 2 + 1 x ) d x \int_{0}^{2} (5x^2 + 1x)\,dx ∫ 0 2 ( 5 x 2 + 1 x ) d x . Uses Definite Integral
Evaluate ∫ 2 6 ( 5 x 2 + 4 x ) d x \int_{2}^{6} (5x^2 + 4x)\,dx ∫ 2 6 ( 5 x 2 + 4 x ) d x . Uses Definite Integral
Evaluate ∫ 2 4 ( 5 x 2 + 2 x ) d x \int_{2}^{4} (5x^2 + 2x)\,dx ∫ 2 4 ( 5 x 2 + 2 x ) d x . Uses Definite Integral
Evaluate ∫ 1 5 ( 6 x 2 + 2 x ) d x \int_{1}^{5} (6x^2 + 2x)\,dx ∫ 1 5 ( 6 x 2 + 2 x ) d x . Uses Definite Integral
Evaluate ∫ 0 4 ( 5 x 2 + 1 x ) d x \int_{0}^{4} (5x^2 + 1x)\,dx ∫ 0 4 ( 5 x 2 + 1 x ) d x . Uses Definite Integral
Evaluate ∫ 2 7 ( 2 x 2 + 5 x ) d x \int_{2}^{7} (2x^2 + 5x)\,dx ∫ 2 7 ( 2 x 2 + 5 x ) d x . Uses Definite Integral
Evaluate ∫ 1 6 ( 4 x 2 + 5 x ) d x \int_{1}^{6} (4x^2 + 5x)\,dx ∫ 1 6 ( 4 x 2 + 5 x ) d x . Uses Definite Integral
Evaluate ∫ 2 7 ( 5 x 2 + 1 x ) d x \int_{2}^{7} (5x^2 + 1x)\,dx ∫ 2 7 ( 5 x 2 + 1 x ) d x . Uses Definite Integral
Evaluate ∫ 1 4 ( 6 x 2 + 2 x ) d x \int_{1}^{4} (6x^2 + 2x)\,dx ∫ 1 4 ( 6 x 2 + 2 x ) d x . Uses Definite Integral
Evaluate ∫ 0 3 ( 6 x 2 + 6 x ) d x \int_{0}^{3} (6x^2 + 6x)\,dx ∫ 0 3 ( 6 x 2 + 6 x ) d x . Uses Definite Integral
Evaluate ∫ 1 6 ( 5 x 2 + 1 x ) d x \int_{1}^{6} (5x^2 + 1x)\,dx ∫ 1 6 ( 5 x 2 + 1 x ) d x . Uses Definite Integral
Evaluate ∫ 1 4 ( 5 x 2 + 2 x ) d x \int_{1}^{4} (5x^2 + 2x)\,dx ∫ 1 4 ( 5 x 2 + 2 x ) d x . Uses Definite Integral
Evaluate ∫ 0 4 ( 6 x 2 + 6 x ) d x \int_{0}^{4} (6x^2 + 6x)\,dx ∫ 0 4 ( 6 x 2 + 6 x ) d x . Uses Definite Integral
Evaluate ∫ 2 7 ( 2 x 2 + 4 x ) d x \int_{2}^{7} (2x^2 + 4x)\,dx ∫ 2 7 ( 2 x 2 + 4 x ) d x . Uses Definite Integral
Evaluate ∫ 1 3 ( 5 x 2 + 3 x ) d x \int_{1}^{3} (5x^2 + 3x)\,dx ∫ 1 3 ( 5 x 2 + 3 x ) d x . Uses Definite Integral
Evaluate ∫ 1 5 ( 3 x 2 + 3 x ) d x \int_{1}^{5} (3x^2 + 3x)\,dx ∫ 1 5 ( 3 x 2 + 3 x ) d x . Uses Definite Integral
Evaluate ∫ 0 3 ( 2 x 2 + 1 x ) d x \int_{0}^{3} (2x^2 + 1x)\,dx ∫ 0 3 ( 2 x 2 + 1 x ) d x . Uses Definite Integral
Evaluate ∫ 0 3 ( 5 x 2 + 2 x ) d x \int_{0}^{3} (5x^2 + 2x)\,dx ∫ 0 3 ( 5 x 2 + 2 x ) d x . Uses Definite Integral
Evaluate ∫ 0 1 x e 1 x d x \int_{0}^{1} x\,e^{1x}\,dx ∫ 0 1 x e 1 x d x using integration by parts. Uses Integration by Parts
Evaluate ∫ 0 1 x e 3 x d x \int_{0}^{1} x\,e^{3x}\,dx ∫ 0 1 x e 3 x d x using integration by parts. Uses Integration by Parts
Evaluate ∫ 0 1 x e 2 x d x \int_{0}^{1} x\,e^{2x}\,dx ∫ 0 1 x e 2 x d x using integration by parts. Uses Integration by Parts
For the quadratic x 2 + 6 x + 8 = 0 x^2 + 6x + 8 = 0 x 2 + 6 x + 8 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 − 3 x − 18 = 0 x^2 - 3x - 18 = 0 x 2 − 3 x − 18 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 1 x − 20 = 0 x^2 + 1x - 20 = 0 x 2 + 1 x − 20 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 4 x − 5 = 0 x^2 + 4x - 5 = 0 x 2 + 4 x − 5 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 − 7 x + 12 = 0 x^2 - 7x + 12 = 0 x 2 − 7 x + 12 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 9 x + 18 = 0 x^2 + 9x + 18 = 0 x 2 + 9 x + 18 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 7 x + 6 = 0 x^2 + 7x + 6 = 0 x 2 + 7 x + 6 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 1 x − 12 = 0 x^2 + 1x - 12 = 0 x 2 + 1 x − 12 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 9 x + 20 = 0 x^2 + 9x + 20 = 0 x 2 + 9 x + 20 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 5 x − 6 = 0 x^2 + 5x - 6 = 0 x 2 + 5 x − 6 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 5 x + 0 = 0 x^2 + 5x + 0 = 0 x 2 + 5 x + 0 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 4 x − 12 = 0 x^2 + 4x - 12 = 0 x 2 + 4 x − 12 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 1 x − 2 = 0 x^2 + 1x - 2 = 0 x 2 + 1 x − 2 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 1 x − 30 = 0 x^2 + 1x - 30 = 0 x 2 + 1 x − 30 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 − 3 x + 2 = 0 x^2 - 3x + 2 = 0 x 2 − 3 x + 2 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 0 x − 9 = 0 x^2 + 0x - 9 = 0 x 2 + 0 x − 9 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 2 x − 15 = 0 x^2 + 2x - 15 = 0 x 2 + 2 x − 15 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 − 13 x + 42 = 0 x^2 - 13x + 42 = 0 x 2 − 13 x + 42 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 7 x + 10 = 0 x^2 + 7x + 10 = 0 x 2 + 7 x + 10 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
For the quadratic x 2 + 6 x + 0 = 0 x^2 + 6x + 0 = 0 x 2 + 6 x + 0 = 0 , find the sum and product of roots, the discriminant, and the nature of the roots. Uses Quadratic Roots (Vieta)
Find the coefficient of x 2 x^{2} x 2 in the expansion of ( 3 x + 1 ) 6 (3x + 1)^{6} ( 3 x + 1 ) 6 . Uses Binomial Theorem
Find the coefficient of x 2 x^{2} x 2 in the expansion of ( 2 x + 1 ) 6 (2x + 1)^{6} ( 2 x + 1 ) 6 . Uses Binomial Theorem
Find the coefficient of x 1 x^{1} x 1 in the expansion of ( 3 x + 1 ) 8 (3x + 1)^{8} ( 3 x + 1 ) 8 . Uses Binomial Theorem
Find the coefficient of x 4 x^{4} x 4 in the expansion of ( 1 x + 1 ) 5 (1x + 1)^{5} ( 1 x + 1 ) 5 . Uses Binomial Theorem
Find the coefficient of x 5 x^{5} x 5 in the expansion of ( 3 x + 1 ) 6 (3x + 1)^{6} ( 3 x + 1 ) 6 . Uses Binomial Theorem