`(i)` `int (ax +b)^n dx = (ax+b)^(n+1)/(a (n+1)) +c` `,n ne -1` `quadquadquadquadquadquadquadquadquadquadquad (ii)` `int dx /(ax +b) =1/a ln (ax +b) +c`

`(iii)` `int e^(ax+b) dx =1/a e^(ax +b) +c` `quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad (iv)` `int a^(px +q) dx =1/p (a^(px+q))/(ln a)` `(a > 0)+c`

`(v)` `int sin (ax +b)dx =-1/a cos (ax +b) +c` `quadquadquadquadquadquad quadquadquadquad (vi)` `int cos (ax+b)dx = 1/a sin (ax+b) +c`

`(vii)` `int tan (ax +b)dx =1/a ln sec (ax+b)+c` `quadquadquadquadquadquadquadquadquad (viii)` `int cot (ax+b)dx =1/a ln sin (ax +b) +c`

`(ix)` `int sec^2 (ax+b) dx =1/a tan (ax +b)+c` `quadquadquadquadquadquad quadquadquad (x)` `int cosec^2 (ax+b)dx =-1/a cot (ax +b)+c`

`(x i)` `int sec(ax + b)* tan(ax + b)dx =1/a sec(ax + b) + c`

`(x ii)` `int cosec (ax + b) * cot (ax + b) dx = -1/a cosec (ax + b) + c`

`(x iii)` `int secx dx = In (secx + tanx) + cquadquad` or `quadquadIn tan ( pi/4 + x/2)+c`

`(x iv)` `int cosec x dx = In (cosec x- cot x) + cquadquad` or `quadquadIn tan (x/2) +cquadquad ` or `quadquad- In (cosccx + cotx)`

`(xv)` `int sinh x dx = cosh x + c`

`(xvi)` `int cosh x dx = sinh x + c`

`(xvii)` `int sech^2x dx = tanh x + c`

`(xviii)` `int cosech^2x dx = - coth x + c`

`(x ix)` `int sech x* tanh x dx = - sech x + c`

`(x x)` `int cosech x * coth x dx = - cosech x + c`

`(x x i)` `int dx /(sqrt(a^2-x^2)) = sin^-1 (x/a) +c`

`(x x ii)` `int dx/(a^2 +x^2)=1/a tan^-1 (x/a) +c`

`(x x iii)` `int dx /(root x (x^2-a^2)) =1/a sec^-1 (x/a) +c`

`(x x iv)` `int dx/sqrt (x^2+a^2) = In [ x+ sqrt (x^2+a^2)]quadquadquad` or `quadquadsin h^-1 (x/a)+c`

`(x xv) ` `int dx /(sqrt (x^2-a^2)) = In [ x + sqrt (x^2-a^2)]quadquadquad` or `quadquadcosh^-1 (x/a) +c`

`(x xvi)` `int dx /(a^2-x^2)=1/(2a) In (a+x)/(a-x)+c`

`(x xvii)` `int dx /(x^2-a^2)=1/(2a) In (x-a)/(x+a) +c`

`(x xviii)` `int sqrt (a^2 -x^2) =x/2 sqrt (a^2 -x^2 )+ a^2/2 sin^-1 (x/a) +c`

`(x x ix)` `int sqrt (x^2 +a^2 )dx =x/2 sqrt (x^2 +a^2) +a^2/2 sin h^-1 (x/a) +c`

`(x x x)` `int sqrt (x^2 -a^2) dx =x/2 sqrt (x^2-a^2) -a^2/2 cos h^-1 (x/a) +c`

`(x x x i)` `int e^(ax) * sin bx dx = (e^(ax))/(a^2+b^2) (a sin bx - b cos bx)+c`

`(x x x ii)` `int e^(ax) * cos bx dx = e^(ax) /(a^2 +b^2)(a cos bx + b sin bx)+c`