Question

integrate 1/√(x^2+4x+10) dx ​

Answer 1

Explanation:

∫ x 2 +4x+105x+3 d5x+3

=A dxd (x 2 +4x+10)+B

⇒5x+3=A(2x+4)+B

⇒5x+3=2Ax+4A+B

on comparing both sides we get

2A=5⇒A= 25

and 4A+B=3⇒4× 2 +B=3⇒B=−7

so, ∫ x 2 +4x+10

5x+3

dx= 25

∫ x 2 +4x+10

(2x+4) dx−7∫ x 2 +4x+10dx

Putting x 2 +4x+10=t

⇒(2x+4)dx=dt

= 25

∫ tdt −7∫ x 2 +4x+4+6dx

= 25 ×2 t −7∫ (x+2) 2 +( 6 ) 2dx

=5 x 2 +4x+10 −7log

x+2+ x 2 +4x+10∣∣ +c ...................

∫ a 2 +x 1 dx=log∣x+ a 2 +x 2 ∣+c