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

1 Answer
Jim H
Mar 30, 2015

int dx / (2sqrt(x) + 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 sqrtx ? Among other things, I know the derivative is 1/(2sqrtx)

I could factor out 2sqrtx 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))dx

Did that help?

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

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

Now, what if I let u=1+sqrtx?

Then du = 1/(2sqrtx) dx and our integral becomes int 1/u du which is lnu, so we can quickly finish:

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

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