User:Xreedl/SandboxForMathsPractice

这个页面用来存放和传递数学题，来自《吉米多维奇》，供某人练习。

不定积分

部分最简积分

$1.\int {\frac {1}{1+x^{2}}}dx={\begin{cases}\arctan x+C\\-\operatorname {arccot} x+C\end{cases}}$

$2.\int {\frac {1}{1-x^{2}}}dx={\frac {1}{2}}\ln \left|{\frac {1+x}{1-x}}\right|+C$

$3.\int {\frac {1}{\sqrt {1-x^{2}}}}dx={\begin{cases}\arcsin x+C\\-\arccos x+C\end{cases}}$

$4.\int {\frac {1}{\sqrt {x^{2}\pm 1}}}dx=\ln \left|x+{\sqrt {x^{2}\pm 1}}\right|+C$

$5.\int a^{x}dx={\frac {a^{x}}{\ln a}}+C(a>0,a\neq 1);\int e^{x}dx=e^{x}+C$

$6.\int {\frac {1}{\sin ^{2}x}}dx=-\cot x+C$

$7.\int {\frac {1}{\cos ^{2}x}}dx=-\tan x+C$

$8.\int \operatorname {sh} xdx=\operatorname {ch} x+C$

$9.\int \operatorname {ch} xdx=\operatorname {sh} x+C$

$10.\int {\frac {1}{\operatorname {sh} ^{2}x}}dx=-\operatorname {coth} x+C$

$11.\int {\frac {1}{\operatorname {ch} ^{2}x}}dx=\operatorname {th} x+C$

• 双曲函数：以下分别为双曲正弦，双曲余弦，双曲正切，双曲余切，双曲正割，双曲余割。它们对于 $x$ 的一切值都有意义（ $\operatorname {cth} x$ 和 $\operatorname {csch} x$ 在 $x=0$ 时无意义，需除外）。

$\operatorname {sh} x={\frac {e^{x}-e^{-x}}{2}}$

$\operatorname {ch} x={\frac {e^{x}+e^{-x}}{2}}$

$\operatorname {th} x={\frac {\operatorname {sh} x}{\operatorname {ch} x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}$

$\operatorname {cth} x={\frac {\operatorname {ch} x}{\operatorname {sh} x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}$

$\operatorname {sech} x={\frac {1}{\operatorname {ch} x}}={\frac {2}{e^{x}+e^{-x}}}$

$\operatorname {csch} x={\frac {1}{\operatorname {sh} x}}={\frac {2}{e^{x}-e^{-x}}}$

2010.09.23

1628. $\int (3-x^{2})^{3}dx$

1629. $\int x^{2}(5-x)^{4}dx$

1630. $\int (1-x)(1-2x)(1-3x)dx$

1631. $\int ({\frac {1-x}{x}})^{2}dx$

1632. $\int ({\frac {a}{x}}+{\frac {a^{2}}{x^{2}}}+{\frac {a^{3}}{x^{3}}})dx$

1633. $\int {\frac {x+1}{\sqrt {x}}}dx$

1634. $\int {\frac {{\sqrt {x}}-2{\sqrt[{3}]{x^{2}}}+1}{\sqrt[{4}]{x}}}dx$

1635. $\int {\frac {(1-x)^{3}}{x{\sqrt[{3}]{x}}}}dx$

1636. $\int (1-{\frac {1}{x^{2}}}){\sqrt {x{\sqrt {x}}}}dx$

1637. $\int {\frac {({\sqrt {2x}}-{\sqrt[{3}]{3x}})^{2}}{x}}dx$

1638. $\int {\frac {\sqrt {x^{4}+x^{-4}+2}}{x^{3}}}dx$

1639. $\int {\frac {x^{2}}{1+x^{2}}}dx$

1640. $\int {\frac {x^{2}}{1-x^{2}}}dx$

1641. $\int {\frac {x^{2}+3}{x^{2}-1}}dx$

1642. $\int {\frac {{\sqrt {1+x^{2}}}+{\sqrt {1-x^{2}}}}{\sqrt {1-x^{4}}}}dx$

2010.09.24

1643. $\int {\frac {{\sqrt {x^{2}+1}}-{\sqrt {x^{2}-1}}}{\sqrt {x^{4}-1}}}dx$

1644. $\int (2^{x}+3^{x})^{2}dx$

1645. $\int {\frac {2^{x+1}-5^{x-1}}{10^{x}}}dx$

1646. $\int {\frac {e^{3x}+1}{e^{x}+1}}dx$

1647. $\int (1+\sin x+\cos x)dx$

1648. $\int {\sqrt {1-\sin 2x}}dx$

1649. $\int \cot ^{2}xdx$

1650. $\int \tan ^{2}xdx$

2010.09.25

1655. $\int {\frac {1}{x+a}}dx$

1656. $\int (2x-3)^{10}dx$

1657. $\int {\sqrt[{3}]{1-3x}}dx$

1658. $\int {\frac {1}{\sqrt {2-5x}}}dx$

1659. $\int {\frac {1}{(5x-2)^{\frac {5}{2}}}}dx$

1660. $\int {\frac {\sqrt[{5}]{1-2x+x^{2}}}{1-x}}dx$

1661. $\int {\frac {1}{2+3x^{2}}}dx$

1662. $\int {\frac {1}{2-3x^{2}}}dx$

1663. $\int {\frac {1}{\sqrt {2-3x^{2}}}}dx$

1664. $\int {\frac {1}{\sqrt {3x^{2}-2}}}dx$

1665. $\int (e^{-x}+e^{-2x})dx$

2010.09.26-27

1666. $\int (\sin 5x-\sin 5\alpha )dx$

1667. $\int {\frac {1}{\sin ^{2}(2x+{\frac {\pi }{4}})}}dx$

1668. $\int {\frac {1}{1+\cos x}}dx$

1669. $\int {\frac {1}{1-\cos x}}dx$

1670. $\int {\frac {1}{1+\sin x}}dx$

(换元)

1674. $\int {\frac {x}{1-x^{2}}}dx$

$\int {\frac {x}{1-x^{2}}}dx=-{\frac {1}{2}}\int {\frac {d(1-x^{2})}{1-x^{2}}}=-{\frac {1}{2}}\int {\frac {dt}{t}}=-{\frac {1}{2}}\ln \left|1-x^{2}\right|+C$

1675. $\int x^{2}{\sqrt[{3}]{1+x^{3}}}dx$

1676. $\int {\frac {x}{3-2x^{2}}}dx$

1677. $\int {\frac {x}{(1+x^{2})^{2}}}dx$

1678. $\int {\frac {x}{4+x^{4}}}dx$

1679. $\int {\frac {x^{3}}{x^{8}-2}}dx$

1680. $\int {\frac {1}{{\sqrt {x}}(1+x)}}dx$

1681. $\int {\frac {1}{x^{2}}}\sin {\frac {1}{x}}dx$

1682. $\int {\frac {1}{x{\sqrt {x^{2}+1}}}}dx$

1683. $\int {\frac {1}{x{\sqrt {x^{2}-1}}}}dx$

1684. $\int {\frac {1}{(x^{2}+1)^{\frac {3}{2}}}}dx$

1685. $\int {\frac {x}{(x^{2}-1)^{\frac {3}{2}}}}dx$

1686. $\int {\frac {x^{2}}{(8x^{3}+27)^{\frac {2}{3}}}}dx$

1687. $\int {\frac {1}{\sqrt {x(1+x)}}}dx$

1688. $\int {\frac {1}{\sqrt {x(1-x)}}}dx$

1689. $\int xe^{-x^{2}}dx$

1690. $\int {\frac {e^{x}}{2+e^{x}}}dx$