Primitives of irrational functions

A≠0 is supposed.

$\ int (ax+b)^ {\ alpha} \, dx= \ frac {1} {(\ alpha+1) has} (ax+b)^ {\ alpha+1} +C$ (α≠-1)

\ int \ frac {1} {\ sqrt {ax^2+bx+c}} \, dx=

= \ left \ {\ begin {matrix} \ frac {1} {\ sqrt {has}} \ operatorname {argsh} \ frac {2ax+b} {\ sqrt {- (b^2-4ac)}} +C & \ rm {\, if \,} & b^2-4ac<0 & \ rm {\, and \,} & a>0 \ \ \ frac {1} {\ sqrt {has}} \ ln|2ax+b| +C & \ rm {\, if \,} & b^2-4ac=0 & \ rm {\, and \,} & a>0 \ \ - \ frac {1} {\ sqrt {- have}} \ operatorname {Arcsin} \ frac {2ax+b} {\ sqrt {b^2-4ac}} +C & \ rm {\, if \,} & b^2-4ac>0 & \ rm {\, and \,} & a<0 \ \ \ end {matrix} \ right.

\ int \ sqrt {ax^2+bx+c} \, dx= \ frac {2ax+b} {4a} \ sqrt {ax^2+bx+c} - \ frac{b^2-4ac} {8a} \ int \ frac {1} {\ sqrt {ax^2+bx+c}} \, dx

\ int \ frac {X} {\ sqrt {ax^2+bx+c}} \, dx= \ frac {\ sqrt {ax^2+bx+c}} {has} - \ frac {B} {2a} \ int \ frac {1} {\ sqrt {ax^2+bx+c}} \, dx

A>0 is supposed

$\ int \ frac {1} {\ sqrt {a^2-x^2}} \, dx= \ operatorname {Arcsin} \ frac {X} {has} +C$

\ int \ frac {1} {\ sqrt {a^2+x^2}} \, dx= \ operatorname {argsh} \ frac {X} {has} +C

\ int \ frac {1} {\ sqrt {x^2-a^2}} \, dx= \ operatorname {argch} \ frac {X} {has} +C

\ int \ sqrt {a^2-x^2} \, dx= \ frac {X} {2} \ sqrt {a^2-x^2} + \ frac {a^2} {2} \ operatorname {Arcsin} \ frac {X} {has} +C

\ int \ sqrt {a^2+x^2} \, dx= \ frac {X} {2} \ sqrt {a^2+x^2} + \ frac {a^2} {2} \ operatorname {argsh} \ frac {X} {has} +C

\ int \ sqrt {x^2-a^2} \, dx= \ frac {X} {2} \ sqrt {x^2-a^2} - \ frac {a^2} {2} \ operatorname {argch} \ frac {X} {has} +C

\ int X \ sqrt {a^2+x^2} \, dx= \ frac {1} {3} \ sqrt {(a^2+x^2) ^3} +C

\ int X \ sqrt {a^2-x^2} \, dx=- \ frac {1} {3} \ sqrt {(a^2-x^2) ^3} +C

\ int X \ sqrt {x^2-a^2} \, dx= \ frac {1} {3} \ sqrt {(x^2-a^2) ^3} +C

\ int \ frac {1} {X} \ sqrt {a^2+x^2} \, dx= \ sqrt {a^2+x^2}- \ ln \ left has|\ frac {1} {X} \ left (a+ \ sqrt {a^2+x^2} \ right) \ right|+C

\ int \ frac {1} {X} \ sqrt {a^2-x^2} \, dx= \ sqrt {a^2-x^2} - has \ ln \ left|\ frac {1} {X} \ left (a+ \ sqrt {a^2-x^2} \ right) \ right|+C

\ int \ frac {1} {X} \ sqrt {x^2-a^2} \, dx= \ sqrt {x^2-a^2} - has \ operatorname {Arccos} \ frac {has} {X} +C

\ int \ frac {X} {\ sqrt {a^2-x^2}} \, dx=- \ sqrt {a^2-x^2} +C

\ int \ frac {X} {\ sqrt {a^2+x^2}} \, dx= \ sqrt {a^2+x^2} +C

\ int \ frac {X} {\ sqrt {x^2-a^2}} \, dx= \ sqrt {x^2-a^2} +C

\int \ frac {x^2} {\ sqrt {a^2-x^2}} \, dx=- \ frac {X} {2} \ sqrt {a^2-x^2} + \ frac {a^2} {2} \ operatorname {Arcsin} \ frac {X} {has} +C

\ int \ frac {x^2} {\ sqrt {a^2+x^2}} \, dx= \ frac {X} {2} \ sqrt {a^2+x^2} - \ frac {a^2} {2} \ operatorname {argsh} \ frac {X} {has} +C

\ int \ frac {x^2} {\ sqrt{x^2-a^2}} \, dx= \ frac {X} {2} \ sqrt {x^2-a^2} + \ frac {a^2} {2} \ operatorname {argch} \ frac {X} {has} +C

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