Integrate

Question

7/(x^2+2x+5)?

Answer 1

Given Function is ⌠ [ 7 / ( x ² + 2x + 5 ) ] dx

Step1 : change denominator form

Take denominator x ² + 2x + 5

x ² + 2x + 1 + 4

x ² + 2(x((1) + 1² + 2²

(x + 1)² + 2²

Step2 :

⌠ [ 7 / ( x ² + 2x + 5 ) ] dx

⌠ [ 7 / ( (x + 1)² + 2² ) ]dx

7⌠ [ 1 / ( (x + 1)² + 2² ) ]dx

Let x+1 = t

Apply derivative

d(x+1) = dt

dx = dt

7⌠ [ 1 / ( t² + 2² ) ]dt

7⌠ [ 1 / ( t² + 2² ) ]dt

Apply formula : ⌠[ 1 / ( x² + a² ) ]dt = (1/a)(tan^-1(x/a)) + c

7 [ (1/2)(tan^-1(t/2)) ] + c

Substitute x+1 = t

(7/2) tan^-1[(x+1)/2] + c

The solution is

⌠ [ 7 / ( x ² + 2x + 5 ) ] dx  =  (7/2) tan^-1[(x+1)/2] + c