Question

$\int \left[f\right(x\left)g^{\prime \prime }\right(x)-f"(x\left)g\right(x\left)\right]dx$ is equal to

• A. $\frac{f\left(x\right)}{g^{\prime }\left(x\right)}$
• B. $f^{\prime }\left(x\right)g\left(x\right)-f\left(x\right)g^{\prime }\left(x\right)$
• C. $f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)$
• D. $f\left(x\right)g^{\prime }\left(x\right)+f^{\prime }\left(x\right)g\left(x\right)$

Answer 1

The correct option is C

$f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)$

$\int \left[f\right(x)g"(x)-f"g(x\left)\right]dx$
$=\int f\left(x\right)g"\left(x\right)dx-\int f"\left(x\right)g\left(x\right)dx$
$=\left(f\right(x\left)g^{\prime }\right(x)-\int f^{\prime }(x\left)g^{\prime }\right(x\left)dx\right)-\left(g\right(x\left)f^{\prime }\right(x)-\int g^{\prime }(x\left)f^{\prime }\right(x\left)dx\right)$
$=f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)$