Integration

Question

∫√(4x²+40x)dx

Answer 1

∫√(4x²+40x)dx

= 2∫√(x²+10x)dx

= 2∫√((x+5)²-25)dx

= 2∫√(u²-25)du , u=x+5

 

This is a standard integral, which you can look up in a table. You find

 

2∫√(u²-25)du = u √(u²-25) -  25 ln (u+√(u²-25)) +c, so that

 

∫√(4x²+40x)dx= (x+5) √((x+5)²-25) -  25 ln (x+5+√((x+5)²-25)) +c

= (x+5) √(x²+10x) -  25 ln (x+5+√(x²+10x)) +c

 

Let me know if you want to see how the formula for the standard integral can be derived.