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Erin Lozano

Erin Lozano

Answered question

2022-06-18

Integral 0 d x x 4 1 x cos ( π ln x ) + 1 + 2 x 2 + 2

Answer & Explanation

pyphekam

pyphekam

Beginner2022-06-19Added 27 answers

Let us introduce the notation
I = 0 d x x 4 1 x cos ( π ln x ) + 1 + 2 x 2 + 2 .
1. Now observe that
1 x 4 1 x cos ( π ln x ) + 1 + 2 x 2 + 2 = 1 x 2 + 1 1 x 2 1 x cos ( π ln x ) + 1 + 2 = = 1 x 2 + 1 cos ( π ln x ) + 1 x x + 1 x + 2 cos ( π ln x ) .
2.Using this formula and making the change of variables x 1 x , we can rewrite I as
I = 0 1 x 2 + 1 cos ( π ln x ) + x x + 1 x + 2 cos ( π ln x ) d x .
3. Summing the last representation with the initial one, we get
2 I = 0 d x 1 + x 2 = π 2 I = π 4 .

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