Find the following integrals. 1. ∫2e^x/e^3x-6x^2x + 11e^x-6 dx

Question

find the following integrals.

1. \(\int \frac{2e^x}{e^{3x}-6e^{2x}+11e^x-6}dx\)

2. \(\int \frac{(3sinx -2)cosx}{5-cos^2x -4sinx} dx\)

Answer 1

1.

⇒ 1 = A(t – 2)(t – 3) + B(t – 1)(t – 3) + C(t – 1)(t – 2)

Put t = 1 ⇒ 1 = A(-1)(-2) ⇒ A = \(\frac{1}{2}\)

Put t = 2 ⇒ 1 = B(1)(-1) ⇒ B = -1

Put t = 3 ⇒ 1 = B(2)(1) ⇒ B = \(\frac{1}{2}\)

2. \(\int \frac{(3sinx -2)cosx}{5-cos^2x -4sinx} dx\)

Put sin x = t ⇒ cosxdx = dt

⇒ 3t – 2 = A(t – 2) + B

Equating the coefficients if t; ⇒ 3 = A

Equating the constants

⇒ -2 = -2A + B ⇒ -2 = -6 + B ⇒ B = 4