Question

if u and v are integrable functions of x then which of the following is true​

Answer 1

Answer:

Let u and v be any two functions of x.

Then, by product rule of differentiation

dx

d

(uv)=u

dx

dv

+v

dx

du

Differentiating both sides with respect to x,

uv=∫u

dx

dv

dx+∫v

dx

du

dx

⇒∫u

dx

dv

dx=uv−∫v

dx

du

dx

Now, put u=f

1

(x)

and

dx

dv

=f

2

(x), i.e., v=∫f

2

(x)dx

∫f

1

(x)f

2

(x)dx=f

1

(x)∫f

2

(x)dx−∫[

dx

d

{f

1

(x)}∫f

2

(x)dx]dx

⇒∫u v dx=u∫v dx−∫[

dx

du

∫vdx]dx

Where, u=f

1

(x),v=f

2

(x).

Here is your answer