Question

Let $g(x)$ be on integrable function and $\int g\left(x\right) dx = g\left(x\right) .$
Statement-1 : $\int g\left(x\right) \left(f \left(x\right) - f ''\left(x\right)\right) dx =g\left(x\right)\left(f \left(x\right) - f '\left(x\right)\right) +C$
Statement-2 : $ \int g\left(x\right) \left(f \left(x\right) + f '\left(x\right)\right) dx = g\left(x\right)f \left(x\right) + C$

• A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement -1
• B. Statement -1 is true, Statement -2 is true ; Statement-2 is NOT a correct explanation for Statement - 1
• C. Statement -1 is false, Statement -2 is true
• D. Statement - 1 is true, Statement- 2 is false

Answer 1

Answer: (B)

$\int g\left(x\right) (f(x)+f'(x))dx $
$\int g\left(x\right) f \left(x\right) dx +\int g\left(x\right)f '\left(x\right)dx$
=$f \left(x\right) \int g\left(x\right)dx-\int\left(f '\left(x\right)\int g\left(x\right) dx\right) dx +\int g\left(x\right) f '\left(x\right)dx$
$=f \left(x\right)g\left(x\right)-\int f '\left(x\right)g\left(x\right)dx+\int g\left(x\right)f '\left(x\right)dx\,\left(\because \int g\left(x\right)dx=g\left(x\right)\right)$
=$f \left(x\right)g\left(x\right)+C$
Now,
$\int g\left(x\right) \left(f \left(x\right)- f ''\left(x\right)\right) dx$
$=\int g\left(x\right)\left\{f \left(x\right)+f '\left(x\right)-\left(f '\left(x\right)+f ''\left(x\right)\right)\right\}dx$
$= g\left(x\right) f\left(x\right) - g\left(x\right) f '\left(x\right) + C$