Xét khai triển:
\(\left(1-x^2\right)^n=C_n^0-C_n^1x^2+C_n^2x^4-...+\left(-1\right)^nC_n^nx^{2n}\)
\(\Leftrightarrow x\left(1-x^2\right)^n=xC_n^0-C_n^1x^3+C_n^2x^4-...+\left(-1\right)^nC_n^nx^{2n+1}\)
Lấy tích phân 2 vế trên đoạn \(\left[0;1\right]\)
\(\int\limits^1_0x\left(1-x^2\right)^ndx=\int\limits^1_0\left(xC_n^0-C_n^1x^3+...+\left(-1\right)^nC_n^nx^{2n+1}\right)\)
\(\Leftrightarrow\dfrac{1}{2\left(n+1\right)}=\dfrac{1}{2}C_n^0-\dfrac{1}{4}C_n^1+\dfrac{1}{6}C_n^2-....+\dfrac{\left(-1\right)^n}{2\left(n+1\right)}C_n^n\)
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