IMPORTANT: For this question, you may use a calculator to compute the final numerical answer after performing integration by parts, but you must not use a calculator to skip the integration process.

Use integration by parts to find the value of the integral

∫
0
1
𝑥
𝑒
𝑎
𝑥
𝑑
𝑥
where a=4

Enter your value rounded to 1 decimal place.

Question

IMPORTANT: For this question, you may use a calculator to compute the final numerical answer after performing integration by parts, but you must not use a calculator to skip the integration process.

Use integration by parts to find the value of the integral

∫
0
1
𝑥
𝑒
𝑎
𝑥
𝑑
𝑥
 where a=4

Enter your value rounded to 1 decimal place.

BlackTom AI · Accepted Answer

Integration by parts: ∫ u dv = uv − ∫ v du.

1. Relevant concepts:
Choose u and dv so that differentiation simplifies u and integration of dv is easy. For integrals of the form x e^{ax}, take u = x (polynomial) and dv = e^{ax} dx (exponential). Also use the antiderivative of e^{ax}, which is (1/a)e^{ax} for constant a ≠ 0.

2. Step-by-step reasoning:
Let u = x → du = dx. Let dv = e^{ax} dx → v = (1/a) e^{ax}.
Apply integration by parts:
∫_0^1 x e^{ax} dx = [u v]_0^1 − ∫_0^1 v du
= [x*(1/a) e^{ax}]_0^1 − ∫_0^1 (1/a) e^{ax} dx.

3. Calculations and intermediate steps:
Evaluate the boundary term:
[x*(1/a) e^{ax}]_0^1 = (1/a) * 1 * e^{a} − (1/a)*0*e^{0} = (1/a) e^{a}.
The remaining integral:
∫_0^1 (1/a) e^{ax} dx = (1/a) * [ (1/a) e^{ax} ]_0^1 = (1/a^2)(e^{a} − 1).
Combine terms:
∫_0^1 x e^{ax} dx = (1/a) e^{a} − (1/a^2)(e^{a} − 1)
= [a e^{a} − (e^{a} − 1)] / a^2
= [e^{a}(a − 1) + 1] / a^2.

Now set a = 4:
e^4 ≈ 54.5981500331.
Compute numerator: 3 * e^4 + 1 = 3*(54.5981500331) + 1 = 163.7944500993 + 1 = 164.7944500993.
Divide by a^2 = 16: 164.7944500993 / 16 ≈ 10.2996531312.
Rounded to one decimal place: 10.3.

4. Why this is correct:
Integration by parts reduced the x factor and used exact antiderivatives of exponentials; algebraic combination yields an exact closed form (3 e^4 + 1)/16, whose decimal evaluation and rounding produce 10.3.

5. Other options:
Not applicable here (short-answer problem). Any alternative valid method (e.g., differentiating a parameterized exponential) yields the same expression; incorrect answers would result from algebra or arithmetic mistakes.

6. Conclusion:
The computed value rounded to one decimal place is 10.3, which matches the provided answer.

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