What is the value of \int_0^1(e^x-x)(e^x+x) {d} x ?

Oxford University MAT 2020 Solved Papers

Question : 2 of 10

Marks: +1, -0

What is the value of

∫↙0^1(e^x-x)(e^x+x) {d} x

Solution:

Using the difference of two squares, this is

∫↙0^1 e^{2 x}-x^2 {d} x

. Notice that, since the derivative of

e^{2 x}

2 e^{2 x}

, this integral is

[\;{1}/{2} e^{2 x}-\;{x^3}/{3}]_0^1=\;{e^2}/{2}-\;{1}/{3}-\;{1}/{2}=\;{3 e^2-5}/{6}

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