Find the following integrals. (i) ∫5x+2/1+2x+3x^2 dx

Question

Find the following integrals.

(i) \(\int \frac{5x+2}{1+2x+3x^2} dx\)

(ii) \(\int \frac{5x+3}{\sqrt{x^2+4x+10}}dx\)

Answer 1

(i) \(\int \frac{5x+2}{1+2x+3x^2} dx\)

Let 5x + 2 = A(6x + 2) + B
⇒ 5 = 6 A ⇒ A = \(\frac{5}{6}\)
⇒ 2 = 2A + B ⇒ 2 = \(\frac{5}{3}\) + B ⇒ 2 – \(\frac{5}{3}\) = \(\frac{1}{3}\)

(ii) \(\int \frac{5x+3}{\sqrt{x^2+4x+10}}dx\)

Let 5x + 3 = A(2x + 4) + B

⇒ 5 = 2A ⇒ A = \(\frac{5}{2}\)

⇒ 3 = 4A + B ⇒ 3 = 10 + B ⇒ B = -7

Using (2) and (3) in (1) we have;