من ويكيبيديا، الموسوعة الحرة
فيما يلي قائمة لتكاملات التوابع غير المنطقة.
توابع تحتوي
[عدل]
![{\displaystyle \int r\;dx={\frac {1}{2}}\left(xr+a^{2}\,\ln \left({\frac {x+r}{a}}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961271c4c25d64fbbd6f3d414bac6f32b17b7c1c)
![{\displaystyle \int r^{3}\;dx={\frac {1}{4}}xr^{3}+{\frac {1}{8}}3a^{2}xr+{\frac {3}{8}}a^{4}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94dfed8d5f78aae70502037ff46d16cafa21f835)
![{\displaystyle \int r^{5}\;dx={\frac {1}{6}}xr^{5}+{\frac {5}{24}}a^{2}xr^{3}+{\frac {5}{16}}a^{4}xr+{\frac {5}{16}}a^{5}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c05cde9b7995d469a8b1af6faecd3fe889089243)
![{\displaystyle \int xr\;dx={\frac {r^{3}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3d78f3e68e641a9b0aab864274113ec4ec57eb)
![{\displaystyle \int xr^{3}\;dx={\frac {r^{5}}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d723e526aaa761351215a402932ce3cf3dd6b73d)
![{\displaystyle \int xr^{2n+1}\;dx={\frac {r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6583ead420aa74655131aaf0ed82cbb82e36157)
![{\displaystyle \int x^{2}r\;dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be3188b42d7d35dd388544a23422b32a0d11aa54)
![{\displaystyle \int x^{2}r^{3}\;dx={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44c78e6b370f74920f2f62974e81ca1da7de0687)
![{\displaystyle \int x^{3}r\;dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c286c93cf9b39345e6c65e118526cf6b0ebba341)
![{\displaystyle \int x^{3}r^{3}\;dx={\frac {r^{3}}{7}}-{\frac {a^{2}r^{5}}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c11195910684997b527e3fab44ef6b7c9088f798)
![{\displaystyle \int x^{3}r^{2n+1}\;dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{3}r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f48c6fc659e5a2eb4a86664f238b542283bc1ee2)
![{\displaystyle \int x^{4}r\;dx={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}}-{\frac {a^{4}xr}{16}}+{\frac {a^{6}}{16}}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36df9dbd1dd669c302c7e4a49233d93311b13c7d)
![{\displaystyle \int x^{4}r^{3}\;dx={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}}-{\frac {a^{4}xr^{3}}{64}}+{\frac {3a^{6}xr}{128}}+{\frac {3a^{8}}{128}}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/297f4516fa8fb6b336558bb5b284d4e07300bca6)
![{\displaystyle \int x^{5}r\;dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbd4bfbe0c7d57a09ac1901c6522cb7e8327d48)
![{\displaystyle \int x^{5}r^{3}\;dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/982e3258c17a21f19d9bdfa66ccbb2e675a32564)
![{\displaystyle \int x^{5}r^{2n+1}\;dx={\frac {r^{2n+7}}{2n+7}}-{\frac {2a^{2}r^{2n+5}}{2n+5}}+{\frac {a^{4}r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e37516c175afbb2ba492a368426ca0a68db63935)
![{\displaystyle \int {\frac {r\;dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\sinh ^{-1}{\frac {a}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa251ebfb074f5c06ca58c44d521a6b1fc1bea28)
![{\displaystyle \int {\frac {r^{3}\;dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a99fe55679d157911cd614e9ca510e54d729d42)
![{\displaystyle \int {\frac {r^{5}\;dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{3}r^{3}}{3}}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e792c993b111fa1cad0e90ab9fddcbf323ccaf)
![{\displaystyle \int {\frac {r^{7}\;dx}{x}}={\frac {r^{7}}{7}}+{\frac {a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0505fdb3e13b7e0704f61b5285649e041ec8d66)
![{\displaystyle \int {\frac {dx}{r}}=\sinh ^{-1}{\frac {x}{a}}=\ln \left|x+r\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7086008adf22252de463c6ba34333d10d2ec3b2)
![{\displaystyle \int {\frac {x\,dx}{r}}=r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a3156a33834a19e192af67f3a54da6ddbe4ce1)
![{\displaystyle \int {\frac {x^{2}\;dx}{r}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\,\sinh ^{-1}{\frac {a}{x}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left|x+r\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ae1fa7fe6d2a5ab668677d6d906c978cd17d8b4)
![{\displaystyle \int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\sinh ^{-1}{\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62b641b02f29a54dfae30532df23f40d76d25a6b)
توابع تحتوي
[عدل]
بفرض أن
، من أجل
، انظر المقطع اللاحق:
![{\displaystyle \int xs\;dx={\frac {1}{3}}s^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ee984a361d44c45466b1c0d5bdddc61425ca6c)
![{\displaystyle \int {\frac {s\;dx}{x}}=s-a\cos ^{-1}\left|{\frac {a}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3796397e7bf4a7e7654dd17d20474cf8bdec98)
![{\displaystyle \int {\frac {dx}{s}}=\int {\frac {dx}{\sqrt {x^{2}-a^{2}}}}=\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0888676523244a1761d956f94a726af0e3617f4d)
لاحظ أن
،
حيث يجب أخذ القيمة الإيجابية لـ
.
![{\displaystyle \int {\frac {x\;dx}{s}}=s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c17011581cfa3d9e53920f319ac77a75013dec77)
![{\displaystyle \int {\frac {x\;dx}{s^{3}}}=-{\frac {1}{s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f2035386db8b8331fafbbb54ba1c1cf994fb5d)
![{\displaystyle \int {\frac {x\;dx}{s^{5}}}=-{\frac {1}{3s^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bcfb12b7156194900353c9def5d1e095149ce99)
![{\displaystyle \int {\frac {x\;dx}{s^{7}}}=-{\frac {1}{5s^{5}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7db234e0dc43716a52f17f53c9b79e30303aaf18)
![{\displaystyle \int {\frac {x\;dx}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/808ffdc762b49c281071fd22b77938afa3e0cd5a)
![{\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}}}=-{\frac {1}{2n-1}}{\frac {x^{2m-1}}{s^{2n-1}}}+{\frac {2m-1}{2n-1}}\int {\frac {x^{2m-2}\;dx}{s^{2n-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819c367c3f01849bfcfa4f310a2e2fc8d4630d2e)
![{\displaystyle \int {\frac {x^{2}\;dx}{s}}={\frac {xs}{2}}+{\frac {a^{2}}{2}}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25191a1c6fa21c48b6b89f476b96e8c7fa54a541)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/755124bfe54faeb5f7029ee29993b42e0335da23)
![{\displaystyle \int {\frac {x^{4}\;dx}{s}}={\frac {x^{3}s}{4}}+{\frac {3}{8}}a^{2}xs+{\frac {3}{8}}a^{4}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95143cd1e023e990284ba780dcf4d8d6089ac254)
![{\displaystyle \int {\frac {x^{4}\;dx}{s^{3}}}={\frac {xs}{2}}-{\frac {a^{2}x}{s}}+{\frac {3}{2}}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2648b5c32f662eb3c1f98576649f188c269aa686)
![{\displaystyle \int {\frac {x^{4}\;dx}{s^{5}}}=-{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}+\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88d29db4a76fe21b51b57e5e47e8136b88280fd5)
![{\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}}}=(-1)^{n-m}{\frac {1}{a^{2(n-m)}}}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1}}{n-m-1 \choose i}{\frac {x^{2(m+i)+1}}{s^{2(m+i)+1}}}\qquad {\mbox{(}}n>m\geq 0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a56fdbdff0c7f9f2437cd5c03b5772496597a6ab)
![{\displaystyle \int {\frac {dx}{s^{3}}}=-{\frac {1}{a^{2}}}{\frac {x}{s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/676ac2308ed60218f4e246884e5783df8e2ebc54)
![{\displaystyle \int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/054a5959ce5e03cf279c1b29dff2ba014ac6dcde)
![{\displaystyle \int {\frac {dx}{s^{7}}}=-{\frac {1}{a^{6}}}\left[{\frac {x}{s}}-{\frac {2}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86843311de7fc72bc01f87742445f7c4b88899e9)
![{\displaystyle \int {\frac {dx}{s^{9}}}={\frac {1}{a^{8}}}\left[{\frac {x}{s}}-{\frac {3}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {3}{5}}{\frac {x^{5}}{s^{5}}}-{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca32b3a8d7f9040840f5d1de3467129edff0d80b)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3}}{3s^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0c92cdcb44ecfe7179711341a1964ef2a0782f)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{7}}}={\frac {1}{a^{4}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96ea4b7b2973dd3e2affa09931a8bf41316161f1)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{9}}}=-{\frac {1}{a^{6}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {2}{5}}{\frac {x^{5}}{s^{5}}}+{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/239ff6c41a3342440c712b9f0c4940e8e6a000d2)
توابع تحتوي
[عدل]
![{\displaystyle \int t\;dx={\frac {1}{2}}\left(xt+a^{2}\sin ^{-1}{\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65b2b9082eb649a91ca680d6203e01461a2e769c)
![{\displaystyle \int xt\;dx=-{\frac {1}{3}}t^{3}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf76bf98582b290fa8da876bde16c9677b5acdc)
![{\displaystyle \int {\frac {t\;dx}{x}}=t-a\ln \left|{\frac {a+t}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad74ec44daf0b93f84e976ed916283da3db60612)
![{\displaystyle \int {\frac {dx}{t}}=\sin ^{-1}{\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33d316558a7b34738d87e5b3c9f124389574df5e)
![{\displaystyle \int {\frac {x^{2}\;dx}{t}}=-{\frac {x}{2}}t+{\frac {a^{2}}{2}}\sin ^{-1}{\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b05d08127a660391aee1351a39fc211b7d6c061)
![{\displaystyle \int t\;dx={\frac {1}{2}}\left(xt-\operatorname {sgn} x\,\cosh ^{-1}\left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(for }}|x|\geq |a|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d22a81b6883c7c7b0326721c3967fadfd239560)
توابع تحتوي
[عدل]
![{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {aR}}+2ax+b\right|\qquad {\mbox{(for }}a>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9dcbcda09d274b680f451515213dee43332fce9)
![{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\,\sinh ^{-1}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f718758e9321f4b67fe5f06f8f029d137eb22d28)
![{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee8134ab3de0e1151a98a0da96aa6a42422a025b)
![{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(for }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5b730269f816739bfcf9db0a522db3a97c76b7)
![{\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}={\frac {4ax+2b}{(4ac-b^{2}){\sqrt {R}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350bd207dd27cf5a13d9cedd433d471f6bf5b10a)
![{\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{5}}}}={\frac {4ax+2b}{3(4ac-b^{2}){\sqrt {R}}}}\left({\frac {1}{R}}+{\frac {8a}{4ac-b^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e98cfb50fe31a8935b8700784450dedb983e8b0)
![{\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{2n+1}}}}={\frac {4ax+2b}{(2n-1)(4ac-b^{2})R^{(2n-1)/2}}}+{\frac {8a(n-1)}{(2n-1)(4ac-b^{2})}}\int {\frac {dx}{R^{(2n-1)/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e917fd2f53afd23bbd44fef4d625ea736c85ff2c)
![{\displaystyle \int {\frac {x\;dx}{\sqrt {ax^{2}+bx+c}}}={\frac {\sqrt {R}}{a}}-{\frac {b}{2a}}\int {\frac {dx}{\sqrt {R}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/600030a37c84403934202860a7eb9ae3b316cd92)
![{\displaystyle \int {\frac {x\;dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}=-{\frac {2bx+4c}{(4ac-b^{2}){\sqrt {R}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac28c9be68dc63f9d05f11b71183336d6997a31)
![{\displaystyle \int {\frac {x\;dx}{\sqrt {(ax^{2}+bx+c)^{2n+1}}}}=-{\frac {1}{(2n-1)aR^{(2n-1)/2}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{(2n+1)/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37e4359d43619c72193e0d13da99da1e0ac6d7cf)
![{\displaystyle \int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {cR}}+bx+2c}{x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25d4bed88b8f638349e32532e1ae1d7917ef85d1)
![{\displaystyle \int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{\sqrt {c}}}\sinh ^{-1}\left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/357577aaeb313751bda3aeb616350c5385b7e61f)
توابع تحتوي
[عدل]
![{\displaystyle \int {\frac {dx}{x{\sqrt {ax+b}}}}\,=\,{\frac {-2}{\sqrt {b}}}\tanh ^{-1}{\sqrt {\frac {ax+b}{b}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95499b4d680cece94732be56d57e74c57ea51ec9)
![{\displaystyle \int {\frac {\sqrt {ax+b}}{x}}\,dx\;=\;2\left({\sqrt {ax+b}}-{\sqrt {b}}\tanh ^{-1}{\sqrt {\frac {ax+b}{b}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ddf991f7c4b497fc370e5bac1c7df2f39022ac)
![{\displaystyle \int {\frac {x^{n}}{\sqrt {ax+b}}}\,dx\;=\;{\frac {2}{a}}\left(x^{n+1}{\sqrt {ax+b}}+bx^{n}{\sqrt {ax+b}}-nb\int {\frac {x^{n-1}}{\sqrt {ax+b}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8607a1f06a843e5411239bf574b4c5595cb66796)
![{\displaystyle \int x^{n}{\sqrt {ax+b}}\,dx\;=\;{\frac {2}{2n+1}}\left(x^{n+1}{\sqrt {ax+b}}+bx^{n}{\sqrt {ax+b}}-nb\int x^{n-1}{\sqrt {ax+b}}\,dx\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8c3bf8206cec454702eaab4cc9d3bf911983b1)
اقرأ أيضاً[عدل]