Respuesta :
Hmm, let's see. To find an equivalent expression for ∫x³eˣdx, we need to apply integration by parts. The correct answer would be option a) 3x²eˣ - ∫ (x⁴/4) eˣdx. This is because when we differentiate 3x²eˣ, we get x³eˣ, which matches the original expression. The remaining integral in option a) is the result of the integration by parts process. So, option a) is the equivalent expression.
Answer: a) 3x²eˣ - ∫ (x⁴/4) eˣdx.
Step-by-step explanation:
Let’s find the equivalent expression for the integral ∫x³eˣdx.
We’ll use integration by parts, which states:
∫udv=uv−∫vdu
Here, let:
(u = x³) (the polynomial part)
(dv = eˣ, dx) (the exponential part)
Now, differentiate (u) to find (du):
du=3x2dx
And integrate (dv) to find (v):
v=∫exdx=ex
Applying the integration by parts formula:
∫x3exdx=x3ex−∫3x2exdx
Now let’s compare this with the given options:
a) (3x²eˣ - \int \frac{{x⁴}}{{4}}eˣ , dx)
This matches our result.
b) (x³eˣ - 3\int x²eˣ , dx)
This is also equivalent.
c) (\int x³ , dx \int eˣ , dx)
This is not correct. The integral of a product is not the product of integrals.
d) (x³\ln(x) - \int x²eˣ \left(\frac{{x⁴}}{{4}}\right) , dx)
This is not equivalent. The term (x³\ln(x)) is not part of our result.
Therefore, the correct equivalent expression is a) 3x²eˣ - ∫ (x⁴/4) eˣdx.
