Using integration by parts how do you solve the definite integral of t^(1/2)* ln t dt from 1 to 4.?

Question

I know you have to do it two times but I can't get the right answer. for the first part i let u = ln t dt and dv = t^1/2 and the second Iet u = 1/t and dv = t ^(3/2). Please show step by step

Answer 1

Uh... the first thing you should notice is that you only have to do it once, not twice. As you mentioned, let u=ln t, du=1/t dt, dv=√t dt, v=2/3 t^(3/2). Then we have:

∫√t ln t dt
2/3 t^(3/2) ln t - 2/3 ∫t^(3/2) * 1/t dt

Notice that t^(3/2) * 1/t = t^(3/2) * t^(-1) = t^(3/2-1) = t^(1/2) = √t. So we have:

2/3 t^(3/2) ln t - 2/3 ∫√t dt

Integrating:

2/3 t^(3/2) ln t - 4/9 t^(3/2)

Now just evaluate at 4 and 1:

(2/3 * 4^(3/2) ln 4 - 4/9 * 4^(3/2)) - (2/3 1^(3/2) ln 1 - 4/9 1^(3/2))

(16/3 ln 4 - 32/9) - (0 - 4/9)
16/3 ln 4 - 28/9

And we are done.

Edit: corrected arithmetical error near end.

Answer 2

there is surely no want for integration via substitution as many answerers have carried out. Integration via areas is plenty much less confusing to do in terms of purposes and derivatives, yet many answerers have chosen the greater complicated way. combine the unique integrand via areas: ? ln?(x + a million) dx = ? ln(x + a million) dx / 2 enable f'(x) = a million f(x) = x enable g(x) = ln(x + a million) g'(x) = a million / (x + a million) ? f'(x)g(x) dx = f(x)g(x) - ? f(x)g'(x) dx ? ln(x + a million) = xln(x + a million) - ? x / (x + a million) dx ? ln(x + a million) = xln(x + a million) - ? (x + a million - a million) / (x + a million) dx ? ln(x + a million) = xln(x + a million) - ? [(x + a million) / (x + a million) - a million / (x + a million)] dx ? ln(x + a million) = xln(x + a million) - ? [a million - a million / (x + a million)] dx ? ln(x + a million) = xln(x + a million) - ? a million dx + ? a million / (x + a million) dx ? ln(x + a million) = xln(x + a million) - x + ln(x + a million) + C ? ln(x + a million) = xln(x + a million) + ln(x + a million) - x + C ? ln(x + a million) = (x + a million)ln(x + a million) - x + C ? ln?(x + a million) dx = [(x + a million)ln(x + a million) - x] / 2 + C

Answer 3

after the first integration by parts
simplify { t ^(3/2) / t } to t^(1/2) and then integrate directly using t^n form

do not go for integration by parts the second time

Answer 4

t^1/2 = du

ln t = v

dt/t = dv

u = 2/3(t^3/2)

uv - Integral u dv

2/3(t^3/2)(ln t) - 2/3(Integral sqrt t dt)

You can solve from here.