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Evaluate ∫ x·eˣ dx, and justify each step.
Integrate by parts, with u = x and dv = eˣ dx. Then du = dx and v = eˣ, so ∫ x·eˣ dx = x·eˣ − ∫ eˣ dx = x·eˣ − eˣ + C.
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Cellular Respiration.pdf
Lecture 7 · 42 pages
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Integration Techniques.pdf
MATH 101 · Lecture 9 · 38 pages
Integration by parts
For a product, ∫u dv = uv − ∫v du. The method works when the new integral is simpler than the one you started with.
Choosing u and dv
Take u to be the polynomial or the logarithm. Differentiating those lowers the degree, so the leftover integral collapses.
SourceIntegration Techniques.pdf · p.14
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41/ 48
85% — review the feedback and worked solutions below.
Correct choice of parts, and the algebra is clean.
Step-by-step solution
With u = x and dv = eˣ dx, du = dx and v = eˣ, so ∫x·eˣ dx = x·eˣ − eˣ + C.
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Professor Whitfield
HIST 210: Modern Europe
17 September 2026
The Marshall Plan Reconsidered
By 1947 western Europe faced a choice between reconstruction and collapse. The plan's architects argued that economic stability was the precondition for any lasting political settlement, and they were prepared to fund it at remarkable cost.
Aid was formally offered to every European state, including those within the Soviet sphere. The offer was as much a diplomatic instrument as an economic one.
.docx · .pptx · .xlsx · .pdf
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