I need help integrating by parts the below
Integral of x^2 (e^(4x) + 3)
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x^2 (e^(4x) + 3)
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Lots of writing here, I suggest you look up the Tabular method
----- diff. -- integ
+ | ... x^2 .. e^4x + 3
- | ... 2x ... (1/4)e^(4x) + 3x
+ | ... 2 .... (1/16)e^(4x) + (3/2)x^2
- | ... 0 ... (1/64)e^(4x) + (1/2)x^3
∫( x^2 (e^(4x) + 3) )dx
= (x^2)((1/4)e^(4x) + 3x) - 2x((1/16)e^(4x) + (3/2)x^2) + 2((1/64)e^(4x) + (1/2)x^3)
= (1/4)e^(4x)(x^2) + 3x^3 - (1/8)x e^4x - 3x^3 + (1/32)e^(4x) + x^3
= (1/32)e^(4x) )(8x^2 - 4x + 1) + x^3
Here is a great video that explains this shortcut
https://www.youtube.com/watch?v=2I-_SV8cwsw&t=695s
To integrate the given function, you can use the technique of integration by parts. The integration by parts formula can be written as follows:
∫u * dv = u * v - ∫v * du
In this case, let's choose u = x^2 and dv = (e^(4x) + 3) dx.
Step 1: Find du (the differential of u)
To find du, differentiate u with respect to x. In this case, we have:
du = d/dx(x^2) dx
Differentiating x^2 gives us:
du = 2x dx
Step 2: Find v (the antiderivative of dv)
To find v, integrate dv. In this case, we have:
v = ∫(e^(4x) + 3) dx
Integrating e^(4x) is a standard integral and gives us:
∫e^(4x) dx = (1/4) * e^(4x)
Integrating 3 gives:
∫3 dx = 3x
So, v is the sum of the two integrals:
v = (1/4) * e^(4x) + 3x
Step 3: Apply the integration by parts formula
Using the integration by parts formula, we have:
∫x^2 (e^(4x) + 3) dx = u * v - ∫v * du
Substituting the values we found in Step 1 and Step 2, we get:
∫x^2 (e^(4x) + 3) dx = x^2 * ((1/4) * e^(4x) + 3x) - ∫((1/4) * e^(4x) + 3x) * (2x dx)
Simplifying, we have:
∫x^2 (e^(4x) + 3) dx = (1/4) * x^2 * e^(4x) + 3x^3 - (1/2) * ∫x^2 * e^(4x) dx - 3 * ∫x^2 dx
Step 4: Solve the remaining integrals
Now we have two integrals left to solve:
∫x^2 * e^(4x) dx and ∫x^2 dx
The integral ∫x^2 dx is a simple power rule integral and can be solved as follows:
∫x^2 dx = (1/3) * x^3
The integral ∫x^2 * e^(4x) dx is more complicated and requires integration by parts again. We can follow the same steps as before to solve this integral.
By applying integration by parts repeatedly on ∫x^2 * e^(4x) dx, you can eventually find the value of the original integral ∫x^2 (e^(4x) + 3) dx.