Class 12 Maths Chapter 7: Integrals MCQs with answers

Answer: (b) sec x + x² + C

d/dx (sec x) = sec x tan x and d/dx (x²) = 2x, so ∫ (sec x tan x + 2x) dx = sec x + x² + C.

Answer: (a) 5^xlog 5 + C

∫ a^x dx = a^xlog a + C; with a = 5 the integral is 5^xlog 5 + C. The power rule does not apply, since the variable is in the exponent.

Answer: (c) 14

F(x) = 2x³ − 2x² + x + C. F(1) = 1 + C = 5 gives C = 4, so F(2) = 16 − 8 + 2 + 4 = 14.

Answer: (c) x(2 log x + 1)

Since ddx ∫ f(x) dx = f(x), f(x) = ddx(x² log x) = 2x log x + x² · 1x = x(2 log x + 1).

Answer: (c) log |cosec x − cot x| + C

Multiplying and dividing by (cosec x − cot x) and putting cosec x − cot x = t gives ∫ cosec x dx = log |cosec x − cot x| + C. Option (d) is ∫ cot x dx; (a) is the derivative of cosec x.

Answer: (b) 15 tan^-1 x5 + C

∫ dxx² + a² = 1a tan^-1 xa + C with a = 5. Option (d) is the form for x² − 25.

Answer: (d) 10

An anti derivative is F(x) = x² + x. F(3) − F(1) = 12 − 2 = 10.

Answer: (d) 2

x⁵ cos x is odd, so its integral over [−1, 1] is 0 by P₇; 3x² is even, so it contributes 2∫₀¹ 3x² dx = 2. The value is 2.

Answer: (a) 8

By the first fundamental theorem, F′(x) = x³ − 2x + 4, so F′(2) = 8 − 4 + 4 = 8. Option (b) is F(2), and (c) differentiates the integrand.

Answer: (b) 14 sin 4x + e^−x + C

ddx(14 sin 4x) = cos 4x and ddx(e^−x) = − e^−x, so the integral is 14 sin 4x + e^−x + C.