Integrate using integration by parts
(integral) (5-x) e^3x
u = 5-x
du = -dx
dv = e^3x
v = 3e^3x
I wonder if this is right so far.
= uv - (integral) v du
= (5-x)(3e^3x) - (integral) (-3e^3x)
=(5-x)(3e^3x) + (integral) (3e^3x)
= (5-x)(3e^3x) + 9e^3x + C
dv = e^3x
v = (1/3)e^3x !!!!!!
Okay then, is this the final answer
= (5-x)(3e^3x) + (integral) -- (1/3)e^3x
= (5-x)(3e^3x) + (1/9)e^3x + C
No. You are using the wrong v function in the u*v term. See Damon's answer.
The (Integral)-v*du term is:
-(Integral)(1/3)e^3x(-dx)
= (1/9)e^3x
So your answer is ok for that term
ok, thank you very much. Happy New Year
x^2
x^2
Your calculation is correct so far. Let's go through the steps of integration by parts to verify:
Given the integral:
∫ (5 - x) e^(3x) dx
Step 1: Choose u and dv
We choose:
u = 5 - x
dv = e^(3x) dx
Step 2: Calculate du and v
To find du, differentiate u with respect to x:
du = d(5 - x)/dx = -1
To find v, integrate dv with respect to x:
v = ∫ e^(3x) dx
= (1/3) e^(3x)
Step 3: Apply the integration by parts formula
Using the formula: ∫ u dv = uv - ∫ v du, we have:
∫ (5 - x) e^(3x) dx = uv - ∫ v du
= (5 - x)(1/3)e^(3x) - ∫ (1/3)e^(3x)(-1) dx
= (5 - x)(1/3)e^(3x) + (1/3) ∫ e^(3x) dx
Step 4: Simplify the integral
Notice that the integral remaining is the same as v. So, we can rewrite the equation as:
∫ (5 - x) e^(3x) dx = (5 - x)(1/3)e^(3x) + (1/3) v
Step 5: Finalize the answer
Evaluating the integral of v, we get:
∫ e^(3x) dx = (1/3) e^(3x)
Therefore, the final answer is:
∫ (5 - x) e^(3x) dx = (5 - x)(1/3)e^(3x) + (1/3) v
= (5 - x)(1/3)e^(3x) + (1/3)(1/3) e^(3x)
= (5 - x)(1/3)e^(3x) + (1/9) e^(3x)
= (5 - x + 1/9)e^(3x) + C
So the fully integrated expression is (5 - x + 1/9)e^(3x) + C, where C is the constant of integration.