题目
计算下列各导数：(1) (d)/(dx)int_(0)^x^2sqrt(1+t^2)dt;(2) (d)/(dx)int_(x^2)^x^3(dt)/(sqrt(1+t^4));(3) (d)/(dx)int_(sin x)^cos xcos(pi t^2)dt.
计算下列各导数：
(1) $\frac{d}{dx}\int_{0}^{x^2}\sqrt{1+t^2}dt$;
(2) $\frac{d}{dx}\int_{x^2}^{x^3}\frac{dt}{\sqrt{1+t^4}}$;
(3) $\frac{d}{dx}\int_{\sin x}^{\cos x}\cos(\pi t^2)dt$.
题目解答
答案
解 (1) $\frac{d}{dx} \int_{0}^{x^2} \sqrt{1+t^2} \, dt = 2x \sqrt{1+x^4}$.
(2) $\frac{d}{dx} \int_{x^2}^{x^3} \frac{dt}{\sqrt{1+t^4}} = \frac{d}{dx} \left( \int_{0}^{x^3} \frac{dt}{\sqrt{1+t^4}} - \int_{0}^{x^2} \frac{dt}{\sqrt{1+t^4}} \right)$
= $\frac{3x^2}{\sqrt{1+x^{12}}} - \frac{2x}{\sqrt{1+x^8}}$.
(3) $\frac{d}{dx} \int_{\sin x}^{\cos x} \cos(\pi t^2) \, dt = \frac{d}{dx} \left[ \int_{0}^{\cos x} \cos(\pi t^2) \, dt - \int_{0}^{\sin x} \cos(\pi t^2) \, dt \right]$
= $-\sin x \cos(\pi \cos^2 x) - \cos x \cos(\pi \sin^2 x)$
= $-\sin x \cos(\pi - \pi \sin^2 x) - \cos x \cos(\pi \sin^2 x)$
= $(\sin x - \cos x) \cos(\pi \sin^2 x).$