Chapter 4 Definite Integration Miscellaneous Exercise 4 Solutions

(a)

(a)

(b) 4 – π

(c)

(d) 4√2

(c) √e(√e − 1)

(a) a = e, b = -2

(d) I1 = I2

(b)

(a) 0

Let I = Put Numerator = A(Denominator) + B[(Denominator)] ∴ cos x = A(3 cos x + sin x) + B[(3 cos x + sin x)] = A(3 cos x + sin x) + B(-3 sin x + cos x) ∴ cos x + 0. sin x = (3A + B) cos x + (A – 3B) sin x Comparing the coefficients of sinx and cos x on both the sides, we get 3A + B = 1 ………. (1) A – 3B = 0 ………. (2) Multiplying equation (1) by 3, we get 9A + 3B = 3 ………(3) Adding (2) and (3), we get 10A = 3 ∴ A =

Let I = Put √x = t ∴ x = t2 and dx = 2t. dt When x = 0, t = 0 When x = 1, t = 1

Let I =

|x – 2|= 2 – x, if x < 2 = x – 2, if x ≥ 2