【微分積分】4-3-3 定積分と不等式｜問題集
1.次の不等式を証明しなさい。
(1)\(\displaystyle \frac{\pi}{4}< \int_0^1\frac{1}{1+x^n}dx< 1\ \ \ (n>2)\)
\(0\leqq x\leqq 1\)のとき、\(1< 1+x^n< 1+x^2\)より
\(\displaystyle \frac{1}{1+x^2}< \frac{1}{1+x^n}< 1\)
積分の単調性より
\(\displaystyle \int_0^1\frac{1}{1+x^2}dx< \int_0^1\frac{1}{1+x^n}dx< \int_0^11dx\)
ここで、それぞれの積分を求めると
\(\displaystyle \frac{\pi}{4}< \int_0^1\frac{1}{1+x^n}dx< 1\)
\(\displaystyle \frac{1}{1+x^2}< \frac{1}{1+x^n}< 1\)
積分の単調性より
\(\displaystyle \int_0^1\frac{1}{1+x^2}dx< \int_0^1\frac{1}{1+x^n}dx< \int_0^11dx\)
ここで、それぞれの積分を求めると
\(\displaystyle \frac{\pi}{4}< \int_0^1\frac{1}{1+x^n}dx< 1\)
(2)\(\displaystyle \frac{1}{2n+2}\leqq \int_0^1\frac{x^n}{1+x}dx\leqq \frac{1}{n}\ \ \ (n\geqq1)\)
\(0\leqq x\leqq 1\)のとき、\(\displaystyle \frac{1}{2}\leqq \frac{1}{1+x}\leqq 1\)より
\(\displaystyle \frac{x^n}{2}\leqq \frac{x^n}{1+x}\leqq x^n\)
積分の単調性より
\(\displaystyle \int_0^1\frac{x^n}{2}dx\leqq \int_0^1\frac{x^n}{1+x}dx\leqq \int_0^1x^ndx\)
ここで、それぞれの積分を求めると
\(\displaystyle \frac{1}{2(n+1)}\leqq \int_0^1\frac{x^n}{1+x}dx\leqq \frac{1}{n+1}\)
\(\displaystyle \frac{1}{n+1}\leqq \frac{1}{n}\)より、
\(\displaystyle \frac{1}{2(n+1)}\leqq \int_0^1\frac{x^n}{1+x}dx\leqq \frac{1}{n}\)
\(\displaystyle \frac{x^n}{2}\leqq \frac{x^n}{1+x}\leqq x^n\)
積分の単調性より
\(\displaystyle \int_0^1\frac{x^n}{2}dx\leqq \int_0^1\frac{x^n}{1+x}dx\leqq \int_0^1x^ndx\)
ここで、それぞれの積分を求めると
\(\displaystyle \frac{1}{2(n+1)}\leqq \int_0^1\frac{x^n}{1+x}dx\leqq \frac{1}{n+1}\)
\(\displaystyle \frac{1}{n+1}\leqq \frac{1}{n}\)より、
\(\displaystyle \frac{1}{2(n+1)}\leqq \int_0^1\frac{x^n}{1+x}dx\leqq \frac{1}{n}\)
次の学習に進もう！