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.