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

Integration of the form f'(x)/(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

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

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)

If int x e ^(2x) is equal to e ^(2x) f (x) + c where C is constant of integration then f (x) is ((2x -1))/( 2)

Integrate : int e^x sin x dx.

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

Read the following text and answer the followig questions on the basis of the same : inte^(x)[f(x) + f'(x)]dx = int e^(x)f(x)dx + int e^(x) f'(x)dx = f(x)e^(x) - int f'(x)e^(x)dx + int f'(x)e^(x)dx = e^(x)f(x) + c int e^(x)(sin x + cos x)dx =