积分日记

最新推荐文章于 2025-07-14 19:03:08 发布

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最新推荐文章于 2025-07-14 19:03:08 发布

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含有 $a x + b$ 的积分

$∫dxax+b=1aln⁡∣ax+b∣+C\int \frac{dx}{ax+b} = \frac{1}{a}\ln |ax+b|+C$

$∫(ax+b)μdx=1a(μ+1)(ax+b)μ+1+C\int (ax+b)^{\mu} dx=\frac{1}{a(\mu+1)}(ax+b)^{\mu+1}+C$

$∫xax+bdx=1a2(ax+b−bln⁡∣ax+b∣)+C\int \frac{x}{ax+b}dx=\frac{1}{a^{2}}(ax+b-b\ln|ax+b|)+C$

$∫x2ax+bdx=1a3[12(ax+b)2−2b(ax+b)+b2ln⁡∣ax+b∣]+C\int \frac{x^{^2}}{ax+b}dx=\frac{1}{a^{3}}[\frac{1}{2}(ax+b)^{2}-2b(ax+b)+b^{2}\ln|ax+b|]+C$

$∫dxx(ax+b)=−1bln⁡∣ax+bx∣+C\int \frac{dx}{x(ax+b)}=-\frac{1}{b}\ln|\frac{ax+b}{x}|+C$

$∫dxx2(ax+b)=−1bx+ab2ln⁡∣ax+bx∣+C\int \frac{dx}{x^{2}(ax+b)}=-\frac{1}{bx}+\frac{a}{b^{2}}\ln|\frac{ax+b}{x}|+C$

$∫x(ax+b)2dx=1a2(ln⁡∣ax+b∣+bax+b)+C\int \frac{x}{(ax+b)^{2}}dx=\frac{1}{a^{2}}(\ln|ax+b|+\frac{b}{ax+b})+C$

$∫x2(ax+b)2dx=1a3(ax+b−2bln⁡∣ax+b∣−b2ax+b)+C\int \frac{x^2}{(ax+b)^2}dx=\frac{1}{a^{3}}(ax+b-2b\ln|ax+b|-\frac{b^{2}}{ax+b})+C$

$∫dxx(ax+b)2=1b(ax+b)−1b2ln⁡∣ax+bx∣+C\int \frac{dx}{x(ax+b)^{2}}=\frac{1}{b(ax+b)}-\frac{1}{b^{2}}\ln|\frac{ax+b}{x}|+C$

含有 $ax+b\sqrt{ax+b}$ 的积分

$∫ax+bdx=23a(ax+b)3+C\int \sqrt{ax+b}dx=\frac{2}{3a}\sqrt{(ax+b)^{3}}+C$

$∫xax+bdx=215a2(3ax−2b)(ax+b)3+C\int x\sqrt{ax+b}dx=\frac{2}{15a^{2}}(3ax-2b)\sqrt{(ax+b)^{3}}+C$

$∫x2ax+bdx=2105a3(15a2x2−12abx+8b2)(ax+b)3+C\int x^{2}\sqrt{ax+b}dx=\frac{2}{105a^{3}}(15a^{2}x^{2}-12abx+8b^{2})\sqrt{(ax+b)^{3}}+C$

$∫xax+bdx=23a2(ax−2b)ax+b+C\int \frac{x}{\sqrt{ax+b}}dx=\frac{2}{3a^{2}}(ax-2b)\sqrt{ax+b}+C$

$∫x2ax+bdx=215a3(3a2x2−4abx+8b2)ax+b+C\int \frac{x^{2}}{\sqrt{ax+b}}dx=\frac{2}{15a^{3}}(3a^{2}x^{2}-4abx+8b^{2})\sqrt{ax+b}+C$

$∫dxxax+b={1bln⁡∣ax+b−bax+b+b∣+C(b>0)2−barctan⁡ax+b−b+C(b<0)\int \frac{dx}{x\sqrt{ax+b}}=\left\{ \begin{aligned} \frac{1}{\sqrt{b}} \ln \left| \frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}} \right| + C \qquad (b>0) \\ \frac{2}{\sqrt{-b}}\arctan \sqrt{\frac{ax+b}{-b}}+C \qquad (b<0) \end{aligned} \right.$

$∫dxx2ax+b=−ax+bbx−a2b∫dxxax+b\int \frac{dx}{x^{2}\sqrt{ax+b}}=-\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}\int \frac{dx}{x\sqrt{ax+b}}$

$∫ax+bxdx=2ax+b+b∫dxxax+b\int \frac{\sqrt{ax+b}}{x}dx=2\sqrt{ax+b}+b\int\frac{dx}{x\sqrt{ax+b}}$

$∫ax+bx2dx=−ax+bx+a2∫dxxax+b\int \frac{\sqrt{ax+b}}{x^{2}}dx=-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}\int \frac{dx}{x\sqrt{ax+b}}$

含有 $x2±a2x^2 \pm a^2$ 的积分

$∫dxx2+a2=1aarctan⁡xa+C\int \frac{dx}{x^2+a^2} = \frac{1}{a} \arctan \frac{x}{a} + C$

$∫dx(x2+a2)n=x2(n−1)a2(x2+a2)n−1+2n−32(n−1)a2∫dx(x2+a2)n−1\int \frac{dx}{(x^2+a^2)^n} = \frac{x}{2(n-1)a^2(x^2+a^2)^{n-1}} + \frac{2n-3}{2(n-1)a^2} \int \frac{dx}{(x^2+a^2)^{n-1}}$

$∫dxx2−a2=12aln⁡∣x−ax+a∣+C\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\ln|\frac{x-a}{x+a}|+C$

含有 $ax2+b(a>0)ax^2+b\quad(a>0)$ 的积分

$∫dxax2+b={1abarctan⁡abx+C(b>0)12−abln⁡∣ax−−bax+−b∣+C(b<0)\int \frac{dx}{ax^2+b}=\left\{ \begin{aligned} \frac{1}{\sqrt{ab}}\arctan\sqrt{\frac{a}{b}}x+C \qquad (b>0)\\ \frac{1}{2\sqrt{-ab}} \ln |\frac{\sqrt{a}x-\sqrt{-b}}{\sqrt{a}x+\sqrt{-b}}|+C \qquad (b<0) \end{aligned} \right.$