Find ∫1/x^2+a^2dx

Question

1. Find ∫\(\frac{1}{x^2+a^2}dx\)

2. Show that 3x + 1 =\(\frac{3}{4}\)(4x - 2) + \(\frac{5}{2}\)

3.  Evaluate \(\int \frac{3x+1}{2x^2+2x + 3}\)dx

Answer 1

1.  ∫\(\frac{1}{x^2+a^2}dx\) = 1/a tan^-1 x/a + c.

2. 3x + 1 = A \(\frac{d}{dx}\)(2x² – 2x + 3) + B

= A(4x – 2) + B

3 = 4A; A = 3/4

1 = -2A + B

1 = -3/2 + B, B = 1 + 3/2 = 5/2

∴ 3x + 1 = 3/4(4x – 2) + 5/2

3.