【微分積分】4-2-2 微分積分の基本公式｜問題集
1.次の関数\(f(t)\)が連続のとき、導関数を求めなさい。
(1)\(\displaystyle \int_x^bf(t)dt\)
\(\displaystyle \frac{d}{dx}\int_x^bf(t)dt\)
\(\displaystyle =-\frac{d}{dx}\int_b^xf(t)dt\)
\(=-f(x)\)
\(\displaystyle =-\frac{d}{dx}\int_b^xf(t)dt\)
\(=-f(x)\)
(2)\(\displaystyle \int_x^{x+1}f(t)dt\)
\(\displaystyle \frac{d}{dx}\int_x^{x+1}f(t)dt\)
\(=f(x+1)-f(x)\)
\(=f(x+1)-f(x)\)
(3)\(\displaystyle \int_0^{2x}x^2f(t)dt\)
\(\displaystyle \frac{d}{dx}\int_0^{2x}x^2f(t)dt\)
\(\displaystyle =\frac{d}{dx}\left(x^2\int_0^{2x}f(t)dt\right)\)
\(\displaystyle =2x\int_0^{2x}f(t)dt+2x^2f(2x)\)
\(\displaystyle =\frac{d}{dx}\left(x^2\int_0^{2x}f(t)dt\right)\)
\(\displaystyle =2x\int_0^{2x}f(t)dt+2x^2f(2x)\)
次の学習に進もう！