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2024 PEA Q7

∫x3ex2 dx\displaystyle\int x^{3}e^{x^{2}}\,dx equals

Reveal answer and solution

Answer

B

Solution

  1. 1

    Let u=x2u=x^{2}, du=2x dxdu=2x\,dx. Then

  2. 2
    ∫x3ex2 dx=∫x2⋅ex2⋅x dx=12∫u eu du=12(u−1)eu+C=x2−12ex2+C. \int x^{3}e^{x^{2}}\,dx =\int x^{2}\cdot e^{x^{2}}\cdot x\,dx =\frac{1}{2}\int u\,e^{u}\,du =\frac{1}{2}(u-1)e^{u}+C =\frac{x^{2}-1}{2}e^{x^{2}}+C.

Answer structure / marking notes

Use ∫ueudu=(u−1)eu\int u e^{u}du=(u-1)e^{u}.

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Content note

Imported from public/resources/isi/msqe/solutions/pea/2024/ISI_MSQE_PEA_2024_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.