Question

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

• 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)

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