Kalpana Kalpana (Editor)

List of integrals of irrational functions

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The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. Throughout this article the constant of integration is omitted for brevity.

Contents

Integrals involving r = √a2 + x2

r d x = 1 2 ( x r + a 2 ln ( x + r ) ) r 3 d x = 1 4 x r 3 + 3 8 a 2 x r + 3 8 a 4 ln ( x + r ) r 5 d x = 1 6 x r 5 + 5 24 a 2 x r 3 + 5 16 a 4 x r + 5 16 a 6 ln ( x + r ) x r d x = r 3 3 x r 3 d x = r 5 5 x r 2 n + 1 d x = r 2 n + 3 2 n + 3 x 2 r d x = x r 3 4 a 2 x r 8 a 4 8 ln ( x + r ) x 2 r 3 d x = x r 5 6 a 2 x r 3 24 a 4 x r 16 a 6 16 ln ( x + r ) x 3 r d x = r 5 5 a 2 r 3 3 x 3 r 3 d x = r 7 7 a 2 r 5 5 x 3 r 2 n + 1 d x = r 2 n + 5 2 n + 5 a 2 r 2 n + 3 2 n + 3 x 4 r d x = x 3 r 3 6 a 2 x r 3 8 + a 4 x r 16 + a 6 16 ln ( x + r ) x 4 r 3 d x = x 3 r 5 8 a 2 x r 5 16 + a 4 x r 3 64 + 3 a 6 x r 128 + 3 a 8 128 ln ( x + r ) x 5 r d x = r 7 7 2 a 2 r 5 5 + a 4 r 3 3 x 5 r 3 d x = r 9 9 2 a 2 r 7 7 + a 4 r 5 5 x 5 r 2 n + 1 d x = r 2 n + 7 2 n + 7 2 a 2 r 2 n + 5 2 n + 5 + a 4 r 2 n + 3 2 n + 3 r d x x = r a ln | a + r x | = r a arsinh a x r 3 d x x = r 3 3 + a 2 r a 3 ln | a + r x | r 5 d x x = r 5 5 + a 2 r 3 3 + a 4 r a 5 ln | a + r x | r 7 d x x = r 7 7 + a 2 r 5 5 + a 4 r 3 3 + a 6 r a 7 ln | a + r x | d x r = arsinh x a = ln ( x + r a ) d x r 3 = x a 2 r x d x r = r x d x r 3 = 1 r x 2 d x r = x 2 r a 2 2 arsinh x a = x 2 r a 2 2 ln ( x + r a ) d x x r = 1 a arsinh a x = 1 a ln | a + r x |

Integrals involving s = √x2 − a2

Assume x2 > a2 (for x2 < a2, see next section):

s d x = 1 2 ( x s a 2 ln ( x + s ) ) x s d x = 1 3 s 3 s d x x = s a arccos | a x | d x s = ln | x + s a |

Here ln | x + s a | = s g n ( x ) arcosh | x a | = 1 2 ln ( x + s x s ) , where the positive value of arcosh | x a | is to be taken.

x d x s = s x d x s 3 = 1 s x d x s 5 = 1 3 s 3 x d x s 7 = 1 5 s 5 x d x s 2 n + 1 = 1 ( 2 n 1 ) s 2 n 1 x 2 m d x s 2 n + 1 = 1 2 n 1 x 2 m 1 s 2 n 1 + 2 m 1 2 n 1 x 2 m 2 d x s 2 n 1 x 2 d x s = x s 2 + a 2 2 ln | x + s a | x 2 d x s 3 = x s + ln | x + s a | x 4 d x s = x 3 s 4 + 3 8 a 2 x s + 3 8 a 4 ln | x + s a | x 4 d x s 3 = x s 2 a 2 x s + 3 2 a 2 ln | x + s a | x 4 d x s 5 = x s 1 3 x 3 s 3 + ln | x + s a | x 2 m d x s 2 n + 1 = ( 1 ) n m 1 a 2 ( n m ) i = 0 n m 1 1 2 ( m + i ) + 1 ( n m 1 i ) x 2 ( m + i ) + 1 s 2 ( m + i ) + 1 ( n > m 0 ) d x s 3 = 1 a 2 x s d x s 5 = 1 a 4 [ x s 1 3 x 3 s 3 ] d x s 7 = 1 a 6 [ x s 2 3 x 3 s 3 + 1 5 x 5 s 5 ] d x s 9 = 1 a 8 [ x s 3 3 x 3 s 3 + 3 5 x 5 s 5 1 7 x 7 s 7 ] x 2 d x s 5 = 1 a 2 x 3 3 s 3 x 2 d x s 7 = 1 a 4 [ 1 3 x 3 s 3 1 5 x 5 s 5 ] x 2 d x s 9 = 1 a 6 [ 1 3 x 3 s 3 2 5 x 5 s 5 + 1 7 x 7 s 7 ]

Integrals involving u = √a2 − x2

u d x = 1 2 ( x u + a 2 arcsin x a ) ( | x | | a | ) x u d x = 1 3 u 3 ( | x | | a | ) x 2 u d x = x 4 u 3 + a 2 8 ( x u + a 2 arcsin x a ) ( | x | | a | ) u d x x = u a ln | a + u x | ( | x | | a | ) d x u = arcsin x a ( | x | | a | ) x 2 d x u = 1 2 ( x u + a 2 arcsin x a ) ( | x | | a | ) u d x = 1 2 ( x u sgn x arcosh | x a | ) (for  | x | | a | ) x u d x = u ( | x | | a | )

Integrals involving R = √ax2 + bx + c

Assume (ax2 + bx + c) cannot be reduced to the following expression (px + q)2 for some p and q.

d x R = 1 a ln | 2 a R + 2 a x + b | (for  a > 0 ) d x R = 1 a arsinh 2 a x + b 4 a c b 2 (for  a > 0 4 a c b 2 > 0 ) d x R = 1 a ln | 2 a x + b | (for  a > 0 4 a c b 2 = 0 ) d x R = 1 a arcsin 2 a x + b b 2 4 a c (for  a < 0 4 a c b 2 < 0 | 2 a x + b | < b 2 4 a c ) d x R 3 = 4 a x + 2 b ( 4 a c b 2 ) R d x R 5 = 4 a x + 2 b 3 ( 4 a c b 2 ) R ( 1 R 2 + 8 a 4 a c b 2 ) d x R 2 n + 1 = 2 ( 2 n 1 ) ( 4 a c b 2 ) ( 2 a x + b R 2 n 1 + 4 a ( n 1 ) d x R 2 n 1 ) x R d x = R a b 2 a d x R x R 3 d x = 2 b x + 4 c ( 4 a c b 2 ) R x R 2 n + 1 d x = 1 ( 2 n 1 ) a R 2 n 1 b 2 a d x R 2 n + 1 d x x R = 1 c ln | 2 c R + b x + 2 c x | ,   c > 0 d x x R = 1 c arsinh ( b x + 2 c | x | 4 a c b 2 ) ,   c < 0 d x x R = 1 c arcsin ( b x + 2 c | x | b 2 4 a c ) ,   c < 0 , b 2 4 a c > 0 d x x R = 2 b x ( a x 2 + b x ) ,   c = 0 x 2 R d x = 2 a x 3 b 4 a 2 R + 3 b 2 4 a c 8 a 2 d x R d x x 2 R = R c x b 2 c d x x R R d x = 2 a x + b 4 a R + 4 a c b 2 8 a d x R x R d x = R 3 3 a b ( 2 a x + b ) 8 a 2 R b ( 4 a c b 2 ) 16 a 2 d x R x 2 R d x = 6 a x 5 b 24 a 2 R 3 + 5 b 2 4 a c 16 a 2 R d x R x d x = R + b 2 d x R + c d x x R R x 2 d x = R x + a d x R + b 2 d x x R x 2 d x R 3 = ( 2 b 2 4 a c ) x + 2 b c a ( 4 a c b 2 ) R + 1 a d x R

Integrals involving S = √ax + b

S d x = 2 S 3 3 a d x S = 2 S a d x x S = { 2 b a r c o t h ( S b ) (for  b > 0 , a x > 0 ) 2 b a r t a n h ( S b ) (for  b > 0 , a x < 0 ) 2 b arctan ( S b ) (for  b < 0 ) S x d x = { 2 ( S b a r c o t h ( S b ) ) (for  b > 0 , a x > 0 ) 2 ( S b a r t a n h ( S b ) ) (for  b > 0 , a x < 0 ) 2 ( S b arctan ( S b ) ) (for  b < 0 ) x n S d x = 2 a ( 2 n + 1 ) ( x n S b n x n 1 S d x ) x n S d x = 2 a ( 2 n + 3 ) ( x n S 3 n b x n 1 S d x ) 1 x n S d x = 1 b ( n 1 ) ( S x n 1 + ( n 3 2 ) a d x x n 1 S )

References

List of integrals of irrational functions Wikipedia


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