Calculus.... I'm not mastered here.. :3
Calculus
Level
3
Evaluate.
ln (2 + √2)/√2
ln (2 + √3)/√3
ln (3 + √3)/√2
ln (2 + √3)/√2
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1 solution
Solution : Let S = 1 − 1/5 − 1/7 + 1/11 + 1/13 − 1/17 − 1/19 + ....
= Σ 1/(12n + 1) − 1/(12n + 5) − 1/(12n + 7) + 1/(12n + 11), where n runs from 0 to infinity.
S = Σ ∫ (1 − x⁴ − x⁶ + x¹⁰)x¹²ⁿ dx, with interval x =[0, 1],
S = ∫ (1 − x⁴)(1 − x⁶)/ (1 − x¹²) dx
S = ∫ (1 − x⁴)(1 − x⁶)/ [(1 − x⁶)(1 + x⁶)] dx
S = ∫ (1 − x⁴)/(1 + x⁶) dx
We will obtain the desire integration is ln(2 + √3)/√3 .