Quinn had $4.00. He spent 1/2 of his money on a baseball card, 1/4 of it on a drink, and 1/8 of it on bubble gum. How much mone
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Let , so that
Then the integral transforms to
Integrate by parts, taking
For 0 < y < 1, we have |1 - y²| = 1 - y², so
It's easy to show that uv approaches 0 as y approaches either 0 or 1, so we just have
Recall the Taylor series for ln(1 + y),
Replacing y with -y² gives the Taylor series
and replacing ln(1 - y²) in the integral with its series representation gives
Interchanging the integral and sum (see Fubini's theorem) gives
Compute the integral:
and we recognize the famous sum (see Basel's problem),
So, the value of our integral is