In my calculus class, we're working on integration by parts specifically, but the worksheet I was given can utilize various methods. Can someone please help me? I'm stuck on a couple of them.
The definite integral from 0 to pi (upper bound= pi, lower bound =0)
of tsin3t dt
The definite integral from 4 to 1 (upper bound= 4, lower bound= 1)
of [sqrt(t)]lnt dt
Thank you so much! I'm just really lost! ><
The definite integral from 0 to pi (upper bound= pi, lower bound =0)
of tsin3t dt
The definite integral from 4 to 1 (upper bound= 4, lower bound= 1)
of [sqrt(t)]lnt dt
Thank you so much! I'm just really lost! ><

t * sin(3t) * dt
u = t
du = dt
dv = sin(3t) * dt
v = (1/3) * cos(3t)
int(v * du) =
uv  int(u * dv) =
(1/3) * t * cos(3t) + (1/3) * int(cos(3t) * dt) =>
(1/3) * t * cos(3t) + (1/3) * (1/3) * sin(3t) + C =>
(1/9) * sin(3t)  (1/3) * t * cos(3t) + C
From 0 to pi
(1/9) * sin(3pi)  (1/3) * pi * cos(3pi)  (1/9) * sin(0) + (1/3) * 0 * cos(0) =>
(1/9) * 0  (1/3) * pi * 1  (1/9) * 0 + 0 =>
(1/3) * pi
sqrt(t) * ln(t) * dt
u = ln(t)
du = dt / t
dv = t^(1/2) * dt
v = (2/3) * t^(3/2)
uv  int(v * du) =>
ln(t) * (2/3) * t^(3/2)  (2/3) * int(t^(3/2) * dt / t) =>
(2/3) * t * sqrt(t) * ln(t)  (2/3) * int(t^(1/2) * dt) =>
(2/3) * t * sqrt(t) * ln(t)  (2/3) * (2/3) * t^(3/2) + C =>
(2/3) * t * sqrt(t) * (ln(t)  (2/3)) + C
from 1 to 4
(2/3) * 4 * sqrt(4) * (ln(4)  (2/3))  (2/3) * 1 * sqrt(1) * (ln(1)  (2/3)) =>
(8/3) * 2 * (ln(4)  (2/3))  (2/3) * (0  (2/3)) =>
(16/3) * (ln(4)  (2/3)) + (4/9) =>
(4/3) * (4 * (ln(4)  (2/3)) + (1/3)) =>
(4/3) * (4ln(4)  (8/3) + (1/3)) =>
(4/3) * (12 * ln(4) / 3  7/3) =>
(4/9) * (12 * 2 * ln(2)  7) =>
(4/9) * (24 * ln(2)  7)
u = t
du = dt
dv = sin(3t) * dt
v = (1/3) * cos(3t)
int(v * du) =
uv  int(u * dv) =
(1/3) * t * cos(3t) + (1/3) * int(cos(3t) * dt) =>
(1/3) * t * cos(3t) + (1/3) * (1/3) * sin(3t) + C =>
(1/9) * sin(3t)  (1/3) * t * cos(3t) + C
From 0 to pi
(1/9) * sin(3pi)  (1/3) * pi * cos(3pi)  (1/9) * sin(0) + (1/3) * 0 * cos(0) =>
(1/9) * 0  (1/3) * pi * 1  (1/9) * 0 + 0 =>
(1/3) * pi
sqrt(t) * ln(t) * dt
u = ln(t)
du = dt / t
dv = t^(1/2) * dt
v = (2/3) * t^(3/2)
uv  int(v * du) =>
ln(t) * (2/3) * t^(3/2)  (2/3) * int(t^(3/2) * dt / t) =>
(2/3) * t * sqrt(t) * ln(t)  (2/3) * int(t^(1/2) * dt) =>
(2/3) * t * sqrt(t) * ln(t)  (2/3) * (2/3) * t^(3/2) + C =>
(2/3) * t * sqrt(t) * (ln(t)  (2/3)) + C
from 1 to 4
(2/3) * 4 * sqrt(4) * (ln(4)  (2/3))  (2/3) * 1 * sqrt(1) * (ln(1)  (2/3)) =>
(8/3) * 2 * (ln(4)  (2/3))  (2/3) * (0  (2/3)) =>
(16/3) * (ln(4)  (2/3)) + (4/9) =>
(4/3) * (4 * (ln(4)  (2/3)) + (1/3)) =>
(4/3) * (4ln(4)  (8/3) + (1/3)) =>
(4/3) * (12 * ln(4) / 3  7/3) =>
(4/9) * (12 * 2 * ln(2)  7) =>
(4/9) * (24 * ln(2)  7)