How to integrate from 4 to 0, (x)/(1+2x)^1/2 dx ?

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Answer 1 · 2017-12-01T12:03:52

# -23/6.#

I hope, the Question is #int_4^0x/(1+2x)^(1/2) dx.#

We use the substitution #(1+2x)^(1/2)=t.#

#:. (1+2x)=t^2 :. 2x=t^2-1, :.2dx=2tdt, or, dx=tdt.#

Also, when #x=4 rArr t=3, and, x=0 rArr t=1.#

#:. I=int_4^0x/(1+2x)^(1/2) dx=1/2int_4^0(2x)/(1+2x)^(1/2)dx,#

#=1/2int_3^1{(t^2-1)/t}tdt=1/2int_3^1(t^2-1)dt,#

#=1/2[t^3/3-t]_3^1,#

#=1/2[(1/3-1)-(9-2)].#

# rArr I=-23/6.#