Integrals – important questions for Boards 2027
CBSE Class 12 Maths
Which is an antiderivative of (3x + 5)/((x + 1)(x + 2))? (A) 2 log|x + 1| + log|x + 2| + C (B) log|x + 1| + 2 log|x + 2| + C (C) 3 log|x + 1| + 5 log|x + 2| + C (D) 2 log|x + 1| − log|x + 2| + C
Answer
(A) 2 log|x + 1| + log|x + 2| + C — Partial fraction decomposition gives coefficients 2 and 1.
Evaluate d/dx [∫(3x² + 2) dx].
Answer
∫(3x² + 2) dx = x³ + 2x + C (d/dx)(x³ + 2x + C) = 3x² + 2 ∴ d/dx [∫(3x² + 2) dx] = 3x² + 2
Using integration by parts, evaluate ∫ x eˣ dx, taking u = x and dv = eˣ dx.
Answer
∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C ∴ ∫ x eˣ dx = eˣ(x − 1) + C
Evaluate ∫cos²x dx using the identity cos 2x = 2cos²x − 1.
Answer
cos 2x = 2cos²x − 1 cos²x = ½(1 + cos 2x) ∫cos²x dx = ½∫(1 + cos 2x) dx = ½(x + ½sin 2x) + C ∴ ∫cos²x dx = x/2 + sin 2x/4 + C
Evaluate ∫₀¹ 2x(x² + 1)³ dx by substitution. First work without limits, integrate in the new variable, substitute back, and then use the original limits.
Answer
Let t = x² + 1, so dt = 2x dx. ∫ 2x(x² + 1)³ dx = ∫ t³ dt = t⁴/4 = (x² + 1)⁴/4 ∫₀¹ 2x(x² + 1)³ dx = [(x² + 1)⁴/4]₀¹ = (2⁴ − 1⁴)/4 ∴ ∫₀¹ 2x(x² + 1)³ dx = 15/4
Evaluate ∫ (3x + 4)/(x² + 2x + 5) dx. Express the numerator as A times the derivative of the denominator plus B, determine A and B by equating coefficients, and then integrate using standard forms.
Answer locked. Unlock all answers on SmartKalike
In the integral ∫ 2x/(x⁴ + 3x² + 2) dx, which substitution converts the integrand and differential into a rational expression in y? (A) y = x² (B) y = x³ (C) y = 1/x (D) y = x⁴
Answer locked. Unlock all answers on SmartKalike
Evaluate ∫[d/dx (x³ − 4x)] dx.
Answer locked. Unlock all answers on SmartKalike
Evaluate ∫ x cos x dx by taking x as the first function and cos x as the second function. Show the integral involving the derivative of the first function that is subtracted.
Answer locked. Unlock all answers on SmartKalike
Evaluate ∫sin³x dx using the identity sin 3x = 3sin x − 4sin³x.
Answer locked. Unlock all answers on SmartKalike
48 more probable questions in this chapter, every one with a full answer.
Unlock all answers on SmartKalike