Primitives de fonctions irrationnelles

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Cet article dresse une liste non exhaustive de primitives de fonctions irrationnelles.

On suppose a ≠ 0 {\displaystyle a\neq 0} .

∫ ( a x + b ) α d x = 1 ( α + 1 ) a ( a x + b ) α + 1 + C {\displaystyle \int (ax+b)^{\alpha }\,\mathrm {d} x={\frac {1}{(\alpha +1)a}}(ax+b)^{\alpha +1}+C} ( α ≠ − 1 {\displaystyle \alpha \neq -1} )


∫ 1 a x 2 + b x + c d x {\displaystyle \int {\frac {1}{\sqrt {ax^{2}+bx+c}}}\,\mathrm {d} x} = { 1 a arsinh ⁡ 2 a x + b − ( b 2 − 4 a c ) + C si  b 2 − 4 a c < 0  et  a > 0 1 a ln ⁡ | 2 a x + b | + C si  b 2 − 4 a c = 0  et  a > 0 − 1 − a arcsin ⁡ 2 a x + b b 2 − 4 a c + C si  b 2 − 4 a c > 0  et  a < 0 {\displaystyle ={\begin{cases}{\frac {1}{\sqrt {a}}}\operatorname {arsinh} {\frac {2ax+b}{\sqrt {-(b^{2}-4ac)}}}+C&{\text{si }}b^{2}-4ac<0{\text{ et }}a>0\\{\frac {1}{\sqrt {a}}}\ln |2ax+b|+C&{\text{si }}b^{2}-4ac=0{\text{ et }}a>0\\-{\frac {1}{\sqrt {-a}}}\operatorname {arcsin} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}+C&{\text{si }}b^{2}-4ac>0{\text{ et }}a<0\\\end{cases}}}


∫ a x 2 + b x + c d x = 2 a x + b 4 a a x 2 + b x + c − b 2 − 4 a c 8 a ∫ 1 a x 2 + b x + c d x {\displaystyle \int {\sqrt {ax^{2}+bx+c}}\,\mathrm {d} x={\frac {2ax+b}{4a}}{\sqrt {ax^{2}+bx+c}}-{\frac {b^{2}-4ac}{8a}}\int {\frac {1}{\sqrt {ax^{2}+bx+c}}}\,\mathrm {d} x}


∫ x a x 2 + b x + c d x = a x 2 + b x + c a − b 2 a ∫ 1 a x 2 + b x + c d x {\displaystyle \int {\frac {x}{\sqrt {ax^{2}+bx+c}}}\,\mathrm {d} x={\frac {\sqrt {ax^{2}+bx+c}}{a}}-{\frac {b}{2a}}\int {\frac {1}{\sqrt {ax^{2}+bx+c}}}\,\mathrm {d} x}


On suppose a > 0 {\displaystyle a>0}

∫ 1 a 2 − x 2 d x = arcsin ⁡ x a + C {\displaystyle \int {\frac {1}{\sqrt {a^{2}-x^{2}}}}\,\mathrm {d} x=\operatorname {arcsin} {\frac {x}{a}}+C}


∫ 1 a 2 + x 2 d x = arsinh ⁡ x a + C {\displaystyle \int {\frac {1}{\sqrt {a^{2}+x^{2}}}}\,\mathrm {d} x=\operatorname {arsinh} {\frac {x}{a}}+C}


∫ 1 x 2 − a 2 d x = arcosh ⁡ x a + C {\displaystyle \int {\frac {1}{\sqrt {x^{2}-a^{2}}}}\,\mathrm {d} x=\operatorname {arcosh} {\frac {x}{a}}+C}


∫ a 2 − x 2 d x = x 2 a 2 − x 2 + a 2 2 arcsin ⁡ x a + C {\displaystyle \int {\sqrt {a^{2}-x^{2}}}\,\mathrm {d} x={\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\operatorname {arcsin} {\frac {x}{a}}+C}


∫ a 2 + x 2 d x = x 2 a 2 + x 2 + a 2 2 arsinh ⁡ x a + C {\displaystyle \int {\sqrt {a^{2}+x^{2}}}\,\mathrm {d} x={\frac {x}{2}}{\sqrt {a^{2}+x^{2}}}+{\frac {a^{2}}{2}}\operatorname {arsinh} {\frac {x}{a}}+C}


∫ x 2 − a 2 d x = x 2 x 2 − a 2 − a 2 2 arcosh ⁡ x a + C {\displaystyle \int {\sqrt {x^{2}-a^{2}}}\,\mathrm {d} x={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}-{\frac {a^{2}}{2}}\operatorname {arcosh} {\frac {x}{a}}+C}


∫ x a 2 + x 2 d x = 1 3 ( a 2 + x 2 ) 3 + C {\displaystyle \int x{\sqrt {a^{2}+x^{2}}}\,\mathrm {d} x={\frac {1}{3}}{\sqrt {(a^{2}+x^{2})^{3}}}+C}


∫ x a 2 − x 2 d x = − 1 3 ( a 2 − x 2 ) 3 + C {\displaystyle \int x{\sqrt {a^{2}-x^{2}}}\,\mathrm {d} x=-{\frac {1}{3}}{\sqrt {(a^{2}-x^{2})^{3}}}+C}


∫ x x 2 − a 2 d x = 1 3 ( x 2 − a 2 ) 3 + C {\displaystyle \int x{\sqrt {x^{2}-a^{2}}}\,\mathrm {d} x={\frac {1}{3}}{\sqrt {(x^{2}-a^{2})^{3}}}+C}


∫ 1 x a 2 + x 2 d x = a 2 + x 2 − a ln ⁡ | 1 x ( a + a 2 + x 2 ) | + C {\displaystyle \int {\frac {1}{x}}{\sqrt {a^{2}+x^{2}}}\,\mathrm {d} x={\sqrt {a^{2}+x^{2}}}-a\ln \left|{\frac {1}{x}}\left(a+{\sqrt {a^{2}+x^{2}}}\right)\right|+C}


∫ 1 x a 2 − x 2 d x = a 2 − x 2 − a ln ⁡ | 1 x ( a + a 2 − x 2 ) | + C {\displaystyle \int {\frac {1}{x}}{\sqrt {a^{2}-x^{2}}}\,\mathrm {d} x={\sqrt {a^{2}-x^{2}}}-a\ln \left|{\frac {1}{x}}\left(a+{\sqrt {a^{2}-x^{2}}}\right)\right|+C}


∫ 1 x x 2 − a 2 d x = x 2 − a 2 − a arccos ⁡ a x + C {\displaystyle \int {\frac {1}{x}}{\sqrt {x^{2}-a^{2}}}\,\mathrm {d} x={\sqrt {x^{2}-a^{2}}}-a\operatorname {arccos} {\frac {a}{x}}+C}


∫ x a 2 − x 2 d x = − a 2 − x 2 + C {\displaystyle \int {\frac {x}{\sqrt {a^{2}-x^{2}}}}\,\mathrm {d} x=-{\sqrt {a^{2}-x^{2}}}+C}


∫ x a 2 + x 2 d x = a 2 + x 2 + C {\displaystyle \int {\frac {x}{\sqrt {a^{2}+x^{2}}}}\,\mathrm {d} x={\sqrt {a^{2}+x^{2}}}+C}


∫ x x 2 − a 2 d x = x 2 − a 2 + C {\displaystyle \int {\frac {x}{\sqrt {x^{2}-a^{2}}}}\,\mathrm {d} x={\sqrt {x^{2}-a^{2}}}+C}


∫ x 2 a 2 − x 2 d x = − x 2 a 2 − x 2 + a 2 2 arcsin ⁡ x a + C {\displaystyle \int {\frac {x^{2}}{\sqrt {a^{2}-x^{2}}}}\,\mathrm {d} x=-{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\operatorname {arcsin} {\frac {x}{a}}+C}


∫ x 2 a 2 + x 2 d x = x 2 a 2 + x 2 − a 2 2 arsinh ⁡ x a + C {\displaystyle \int {\frac {x^{2}}{\sqrt {a^{2}+x^{2}}}}\,\mathrm {d} x={\frac {x}{2}}{\sqrt {a^{2}+x^{2}}}-{\frac {a^{2}}{2}}\operatorname {arsinh} {\frac {x}{a}}+C}


∫ x 2 x 2 − a 2 d x = x 2 x 2 − a 2 + a 2 2 arcosh ⁡ x a + C {\displaystyle \int {\frac {x^{2}}{\sqrt {x^{2}-a^{2}}}}\,\mathrm {d} x={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}+{\frac {a^{2}}{2}}\operatorname {arcosh} {\frac {x}{a}}+C}
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