Table of Integrals

Table of Integrals Basic Forms xndx=1n+ 1xn+1(1) 1xdx= ln|x|(2) udv=uv vdu(3) 1ax+bdx=1aln|ax+b|(4) Integrals of Rational Functions 1(x+a)2dx= 1x+a(5) (x+a)ndx=(x+a)n+1n+ 1,n6= 1(6) x(x+a)ndx=(x+a)n+1((n+ 1)x a)(n+ 1)(n+ 2)(7) 11 +x2dx= tan 1x(8) 1a2+x2dx=1atan 1xa(9) xa2+x2dx=12ln|a2+x2|(10) x2a2+x2dx=x atan 1xa(11) x3a2+x2dx=12x2 12a2ln|a2+x2|(12) 1ax2+bx+cdx=2 4ac b2tan 12ax+b 4ac b2(13) 1(x+a)(x+b)dx=1b alna+xb+x, a6=b(14) x(x+a)2dx=aa+x+ ln|a+x|(15) xax2+bx+cdx=12aln|ax2+bx+c| ba 4ac b2tan 12ax+b 4ac b2(16) Integrals with Roots x adx=23(x a)3/2(17) 1 x adx= 2 x a(18) 1 a xdx= 2 a x(19) x x adx=23a(x a)3/2+25(x a)5/2(20) ax+bdx=(2b3a+2x3) ax+b(21) (ax+b)3/2dx=25a(ax+b)5/2(22) x x adx=23(x 2a) x a(23) xa xdx= x(a x) atan 1 x(a x)x a(24) xa+xdx= x(a+x) aln[ x+ x+a](25) x ax+bdx=215a2( 2b2+abx+ 3a2x2) ax+b(26) x(ax+b)dx=14a3/2[(2ax+b) ax(ax+b) b2ln a x+ a(ax+b) ](27) x3(ax+b)dx=[b12a b28a2x+x3] x3(ax+b)+b38a5/2ln a x+ a(ax+b) (28) x2 a2dx=12x x2 a2 12a2ln x+ x2 a2 (29) a2 x2dx=12x a2 x2+12a2tan 1x a2 x2(30) x x2 a2dx=13(x2 a2)3/2(31) 1 x2 a2dx= ln x+ x2 a2 (32) 1 a2 x2dx= sin 1xa(33) x x2 a2dx= x2 a2(34) x a2 x2dx= a2 x2(35) x2 x2 a2dx=12x x2 a2 12a2ln x+ x2 a2 (36) ax2+bx+cdx=b+ 2ax4a ax2+bx+c+4ac b28a3/2ln 2ax+b+ 2 a(ax2+bx+c) (37) x ax2+bx+c=148a5/2(2 a ax2+bx+c ( 3b2+ 2abx+ 8a(c+ax2))+3(b3 4abc) ln b+ 2ax+ 2 a ax2+bx+c )(38) 1 ax2+bx+cdx=1 aln 2ax+b+ 2 a(ax2+bx+c) (39) x ax2+bx+c

Integrals with Trigonometric Functions Z sinaxdx= 1 a cosax (63) Z sin2 axdx= x 2 sin2ax 4a (64) Z sinn axdx= 1 a cosax 2F 1 1 2; 1 n 2; 3 2;cos2 ax (65) Z sin3 axdx= 3cosax 4a …

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