Evaluate ∫ ln ⁡ 2 ln ⁡ 4 e − x 1 − e −...

Finley Mckinney

Finley Mckinney

Answered

2022-06-19

Evaluate
ln 2 ln 4 e x 1 e 2 x d x

Answer & Explanation

Nola Rivera

Nola Rivera

Expert

2022-06-20Added 21 answers

Observe that
1 / 4 1 / 2 d u 1 u 2 = arcsin y | 1 / 4 1 / 2 = arcsin 1 2 arcsin 1 4
But arcsin x = arctan x 1 x 2 , so the above result equals
arcsin 1 2 arcsin 1 4 = arctan 1 3 arctan 1 15 = π 6 arctan 1 15
which equals arcsin 15 4 π 3
Yahir Tucker

Yahir Tucker

Expert

2022-06-21Added 8 answers

Notice that :
1 / 4 1 / 2 d u 1 u 2 = [ cos 1 u ] 1 / 4 1 / 2 = cos 1 1 4 cos 1 1 2 = cos 1 1 4 π 3
But we know that :
cos 1 x = sin 1 1 x 2
then :
1 / 4 1 / 2 d u 1 u 2 = sin 1 1 1 16 π 3 = sin 1 15 4 π 3

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