Can someone help me to break down this integral completely s

Can someone help me to break down this integral completely, specifically how the chain rule applies in the integration? Thank you

Take 4r^2 + 1 = t
differentiating
8r dr = dt

so we have dr = dt/8r

so first integration can be written as

int(sqrt(t) * r * dt/8r)

=> int(sqrt(t) * dt/8)

solving we get

=> (1/12)t^(3/2)

Substituting for t

(1/12)(4r^2 + 1)^(3/2)

Putting the limits from 0 to sqrt(3) we get

(1/12) * 13^(3/2) - (1/12)*1 = (1/12)*(13sqrt(13) - 1)

Now for the second integration we get

int((1/12)*(13sqrt(13) - 1) * d)

we get

(1/12)*(13sqrt(13) - 1) *

Putting limits from 0 to 2 we get

(1/12)*(13sqrt(13) - 1) * 2

Final answer (/6)*(13sqrt(13) - 1)