Question

can anyone answer this question​

Answer 1

Answer:

Answer is e^x f(x) + C

it's a property

Answer 2

 ∫eˣ(f(x) + f'(x))dx  = eˣ f(x)   + C

Step-by-step explanation:

 ∫eˣ(f(x) + f'(x))dx  = eˣ f(x)   + C

 ∫eˣ(f(x) + f'(x))dx

=  ∫eˣ(f(x)dx  + ∫eˣ f'(x)dx

∫udv  = uv - ∫vdu  + C

u = eˣ    =>   du = eˣ dx

dv =  f'(x)

v = f(x)

=   ∫eˣ(f(x)dx + eˣf(x)  -  ∫ f(x)eˣdx  + C

=  eˣf(x) + ∫eˣ(f(x)dx -  ∫ eˣf(x)dx + C

= eˣf(x) + C

Verification

y  =  eˣf(x) + C

=> dy/dx  =  eˣf'(x) +  eˣf(x)

=> dy/dx  =  eˣ(f'(x) +  f(x))

=> dy/dx  =  eˣ(f(x) +  f'(x))

=> y  = ∫  ∫eˣ(f(x) + f'(x))dx

 ∫eˣ(f(x) + f'(x))dx  = eˣ f(x)   + C

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