How do you integrate dx / (2sqrt(x) + 2xdx2x+2x?

1 Answer
Mar 30, 2015

int dx / (2sqrt(x) + 2x )dx2x+2x

Think about integration techniques, think about derivatives, think about fractions and factoring and . . .

There are "obvious" common factors in the denominator.

What do I know about sqrtxx ? Among other things, I know the derivative is 1/(2sqrtx)12x

I could factor out 2sqrtx2x in the denominator. Will that help?
Try it and see. (Yes, I can try it in my head. Students will need to try it on paper.)

int dx / (2sqrt(x) + 2x ) = int 1/(2sqrtx(1+sqrtx))dxdx2x+2x=12x(1+x)dx

Did that help?

Not sure? write it as a product of 2 fracions where one fraction is the derivative of sqrtxx

int dx / (2sqrt(x) + 2x ) = int 1/(2sqrtx) 1/(1+sqrtx)dxdx2x+2x=12x11+xdx

Now, what if I let u=1+sqrtxu=1+x?

Then du = 1/(2sqrtx) dxdu=12xdx and our integral becomes int 1/u du1udu which is lnulnu, so we can quickly finish:

int dx / (2sqrt(x) + 2x ) = ln(1+sqrtx) +Cdx2x+2x=ln(1+x)+C