can you help me get started on this integral by parts?

4
S sqrt(t) ln(t) dt
1

please help! thanks!

Integral t^(1/2)Ln(t)dt =

2/3 t^(3/2)Ln(t)-
2/3 Integral t^(1/2) dt =

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

Simpler method:

Integral t^(a)dt = t^(a+1)/(a+1)

Integral d/da [t^(a)]dt =
d/da [t^(a+1)/(a+1)] ----->

Integral t^(a)Ln(t)dt =

t^(a+1)Ln(t)/(a+1) - t^(a+1)/(a+1)^2

Finally put a = 1/2.

To get started on this integration problem, we can use the method of integration by parts. The formula for integration by parts is:

∫u*v dx = u*∫v dx - ∫u'*(∫v dx) dx

In this case, we can choose u as Ln(t) and dv as sqrt(t) dt.

To find u', we need to differentiate u with respect to t. The derivative of Ln(t) is 1/t, so u' = 1/t.

To find ∫v dx, we need to integrate sqrt(t) with respect to t. The integral of sqrt(t) dt is (2/3) t^(3/2).

Now we can plug these values into the formula:

∫sqrt(t) Ln(t) dt = Ln(t) * (2/3) t^(3/2) - ∫(1/t) * (2/3) t^(3/2) dt

Simplifying the right side:

= (2/3) t^(3/2) Ln(t) - (2/3) ∫t^(1/2) dt

The integral on the right side, ∫t^(1/2) dt, is an easier integral to solve. Using the formula for the integral of a power function:

∫t^(a) dt = t^(a+1)/(a+1)

We can see that when a = 1/2, the integral becomes:

∫t^(1/2) dt = t^(3/2)/(3/2) = 2/3 t^(3/2)

Plugging this back into the original equation:

= (2/3) t^(3/2) Ln(t) - (2/3) * (2/3) t^(3/2)

Simplifying further:

= (2/3) t^(3/2) Ln(t) - 4/9 t^(3/2)

So the answer to the integral ∫sqrt(t) Ln(t) dt is (2/3) t^(3/2) Ln(t) - 4/9 t^(3/2).

To solve the integral ∫(1 to 4) √(t) ln(t) dt using integration by parts, you can follow these steps:

Step 1: Choose u and dv
Let u = ln(t) and dv = √(t) dt.

Step 2: Find du and v
Differentiate u to find du:
du = (1/t) dt.

Integrate dv to find v:
Integrating √(t) dt, we get:
v = (2/3) t^(3/2).

Step 3: Apply the integration by parts formula
Using the integration by parts formula:

∫ u dv = uv - ∫ v du, we have:

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

Simplifying further, we get:

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

Step 4: Evaluate the remaining integral
The remaining integral, ∫ t^(1/2) dt, can be easily integrated using the power rule:

∫ t^(1/2) dt = (2/3) t^(3/2).

Step 5: Finalize the solution
Substituting the values back into the equation from Step 3, we have:

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

= (2/3) t^(3/2) ln(t) - (2/3) (2/3) t^(3/2) + C

= (2/3) t^(3/2) ln(t) - (4/9) t^(3/2) + C.

Therefore, the solution to the integral is:

∫(1 to 4) √(t) ln(t) dt = (2/3) t^(3/2) ln(t) - (4/9) t^(3/2) + C.

Note: In the simpler method mentioned above, we make use of a general property and the power rule to directly find the integral without using integration by parts.