Notes in 07定积分的计算

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Published	10/05/2024	牛顿-莱布尼兹公式如果函数\(F(x)\)是连续函数\(f(x)\)在区间\([a, b]\)上的一个原函数,则\(\displaystyle\int_{a}^{b} f(x) \mathrm{d} x=\text{______} .\)
Published	10/05/2024	定积分换元积分法设 \(f(x)\) 在区间 \([a, b]\) 上连续, 函数 \(x=\varphi(t)\) 满足以下条件:(1) \(\varphi(\alpha)=a, \varphi(\beta)=b\);(2) \(\varphi(t)\) 在 \([\alpha, \beta]\)…
Published	10/05/2024	定积分分部积分法设函数 \(u(x)\) 和 \(v(x)\) 在 \([a, b]\) 上有连续一阶导数, 则\(\displaystyle\int_{a}^{b} u \mathrm{~d} v=\text{______.}\)
Published	10/05/2024	利用奇偶性计算定积分\(\text { (1) 若 } f(-x)=-f(x) \text {, 则 } \displaystyle\int_{-a}^{a} f(x) \mathrm{d} x=\text{______.}\)\(\text { (2) 若 } f(-x)=f(x) \text {…
Published	10/05/2024	利用周期性计算定积分设 \(f(x)\) 是以 \(T\) 为周期的连续函数, 则对任给数 \(a\), 总有\(\displaystyle\displaystyle\int_{a}^{a+T} f(x) \mathrm{d} x=\text{______.}\)
Published	10/05/2024	利用华里士公式计算定积分1\(\displaystyle\int_{0}^{\frac{\pi}{2}} \sin ^{n} x \mathrm{~d} x=\displaystyle\int_{0}^{\frac{\pi}{2}} \cos ^{n} x \mathrm{~d} x\)=\(\te…
Published	10/05/2024	利用华里士公式计算定积分2\(\displaystyle\int_{0}^{\pi} \sin ^{n} x \mathrm{~d} x=\text{______.}\)
Published	10/05/2024	利用华里士公式计算定积分3\(\displaystyle\displaystyle\displaystyle\int_{0}^{\pi} \cos ^{n} x \mathrm{~d} x=\text{______.}\)
Published	10/05/2024	利用华里士公式计算定积分4\(\displaystyle\int_{0}^{2 \pi} \cos ^{n} x \mathrm{~d} x=\displaystyle\int_{0}^{2 \pi} \sin ^{n} x \mathrm{~d} x=\)_____.
Published	10/05/2024	利用区间再现公式计算定积分1设 \(f(x)\) 在 \([a, b]\) 上连续,证明\(\displaystyle\displaystyle\int_{a}^{b} f(x) \mathrm{d} x=\displaystyle\displaystyle\int_{a}^{b} f(a+b-x)…
Published	10/05/2024	利用区间再现公式计算定积分2\(\displaystyle\int_{a}^{b} f(x) \mathrm{d} x=\frac{1}{2} \displaystyle\int_{a}^{b}[f(x)+f(a+b-x)] \mathrm{d} x .\)
Published	10/05/2024	利用区间再现公式计算定积分3\(\displaystyle\displaystyle\int_{a}^{b} f(x) d x=\displaystyle\displaystyle\int_{\frac{a+b}{2}}^{b}[f(x)+f(a+b-x)] d x\)
Published	10/05/2024	利用区间再现公式计算定积分4\(\displaystyle\displaystyle\int_{a}^{b} f(x) \mathrm{d} x=\displaystyle\displaystyle\int_{a}^{\frac{a+b}{2}}[f(x)+f(a+b-x)] \mathrm{d} …
Published	10/05/2024	利用区间再现公式计算定积分5\(\displaystyle\displaystyle\int_{0}^{\pi} x f(\sin x) d x=\frac{\pi}{2} \displaystyle\displaystyle\int_{0}^{\pi} f(\sin x) d x\)
Published	10/05/2024	利用区间再现公式计算定积分6\(\displaystyle\displaystyle\int_{0}^{\pi} x f(\sin x) \mathrm{d} x=\pi \displaystyle\displaystyle\int_{0}^{\frac{\pi}{2}} f(\sin x) \ma…
Published	10/05/2024	利用区间再现公式计算定积分7\(\displaystyle\int_{0}^{\frac{\pi}{2}} f(\sin x) \mathrm{d} x=\displaystyle\int_{0}^{\frac{\pi}{2}} f(\cos x) \mathrm{d} x .\)
Published	10/05/2024	利用区间再现公式计算定积分8\(\displaystyle\int_{0}^{\frac{\pi}{2}} f(\sin x, \cos x) \mathrm{d} x=\displaystyle\int_{0}^{\frac{\pi}{2}} f(\cos x, \sin x) \mathrm{d…
Published	10/05/2024	利用区间再现公式计算定积分9if \(f(x+T)=f(x)\)prove: \(\displaystyle\displaystyle\int_{a}^{a+T} f(x) d x=\displaystyle\displaystyle\int_{b}^{b+T} f(x) d x\)
Published	10/05/2024	2018年数学二真题大题某一步\(I=\displaystyle\displaystyle\int_{0}^{2 \pi}(x-\sin x)(1-\cos x)^{2} d x\)
Published	10/05/2024	利用区间再现公式计算定积分11\(\text { 设 } f(x) \text { 在 }\left[0, \frac{\pi}{2}\right] \text { 上连续, 求 } \displaystyle\displaystyle\int_{0}^{\frac{\pi}{2}} \frac{f…
Published	10/05/2024	利用区间再现公式计算定积分10\(\text { 证明: } \displaystyle\displaystyle\int_{0}^{1} x^{m}(1-x)^{n} \mathrm{~d} x=\displaystyle\displaystyle\int_{0}^{1} x^{n}(1-x)^{…
Published	10/05/2024	设 \(f(x)\) 在 \([0,1]\) 上连续, \(n \in \mathbf{Z}\), 证明\(\displaystyle\displaystyle\int_{\frac{n}{2} \pi}^{\frac{n+1}{2} \pi} f(\|\sin x\|) \mathrm{d} x=\d…
Published	10/05/2024	利用定积分的几何意义计算定积分\(\displaystyle\displaystyle\displaystyle\displaystyle\int_{0}^{a} \sqrt{a^{2}-x^{2}} \mathrm{~d} x=\text{______.}\)\(\displaystyle\dis…
Published	10/05/2024	定积分计算思维脑图
Published	10/05/2024	利用华里士公式计算定积分5【选自武忠祥高数讲义】\(\begin{align}\int_0^\pi x \sin ^n x d x(n \geqslant 1)\end{align}\)_____.
Published	10/05/2024	利用华里士公式推广计算定积分6【了解即可】\(\begin{align}I(m, n)=\int_0^{\frac{\pi}{2}} \cos ^m x \sin ^n x d x\end{align}\)
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