Integration of the form `(f'(x)) / f(x) dx`

Integration of form e^(x)(f(x)+f'(x))dx

Integral of form (f(x))^(n)f'(x)dx

Read the following passages and answer the following questions (7-9) Consider the integrals of the form l=inte^(x)(f(x)+f'(x))dx By product rule considering e^(x)f(x) as first integral and e^(x)f'(x) as second one, we get l=e^(x)f(x)-int(f(x)+f'(x))dx=e^(x)f(x)+c l=inte^(x)(tan^(-1)x+(2x)/((1+x^(2))^(2)))dx then l is equal to

If int(dx)/(sqrt(e^(x)-1))=2tan^(-1)(f(x))+C , (where x gt0 and C is the constant of integration ) then the range of f(x) is

The integral I=inte^(x)((1+sinx)/(1+cosx))dx=e^(x)f(x)+C (where, C is the constant of integration). Then, the range of y=f(x) (for all x in the domain of f(x) ) is

If int(dx)/(x^(2)+x)=ln|f(x)|+C (where C is the constant of integration), then the range of y=f(x), AA x in R-{-1, 0} is

Integration OF ( dx / a Sinx + b Cosx ) type || Integration OF [ e^x( f(x)+f'(x) ] type || Excercise 1 - Q 27, Ex - level 1 ( Q 7)

Let inte^(x).x^(2)dx=f(x)e^(x)+C (where, C is the constant of integration). The range of f(x) as x in R is [a, oo) . The value of (a)/(4) is