Question

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

• A. $\frac{f\prime \left(x\right)}{g\prime \left(x\right)}$
• 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 $f\left(x\right)g\prime \left(x\right)-f\prime \left(x\right)g\left(x\right)$

$\int \left[f\right(x\left)g^{\prime \prime }\right(x)-f^{\prime \prime }(x\left)g\right(x\left)\right]dx$
By parts integration,
$=f\left(x\right)g^{\prime }\left(x\right)-\int f^{\prime }\left(x\right)g^{\prime }\left(x\right)dx-f^{\prime }\left(x\right)g\left(x\right)+\int f^{\prime }\left(x\right)g^{\prime }\left(x\right)dx$
$=f\left(x\right)g^{\prime }\left(x\right)-f^{\prime }\left(x\right)g\left(x\right)$