Question

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

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

Answer 1

The correct option is A $f\left(x\right)g^{{}^{\prime }}\left(x\right)-g\left(x\right)f^{{}^{\prime }}\left(x\right)+c$

$\int \left[fcx\right)g^{\prime \prime }\left(x\right)-f^{\prime \prime }\left(x\right)g\left(x\right)]dx$
$\int f\left(x\right)g^{{}^{\prime \prime }}\left(x\right)+f^{{}^{\prime }}\left(x\right)g^{{}^{\prime }}\left(x\right)dx$
$-\int \left[f^{{}^{\prime }}\right(x\left)g^{{}^{\prime }}\right(x)+f^{{}^{\prime \prime }}(x\left)g\right(x\left)\right]dx$
$\int d\left[fcx\right)g^{{}^{\prime }}\left(x\right)]-\int d(f^{{}^{\prime }}\left(x\right)g\left(x\right)]$
$=f\left(x\right)g^{{}^{\prime }}\left(x\right)-f^{{}^{\prime }}\left(x\right)g\left(x\right)+c$
$\int \left[f\right(x\left)g^{{}^{\prime \prime }}\right(x)-f^{{}^{\prime \prime }}(x\left)g\right(x\left)\right]dx$
$=f\left(x\right)g^{{}^{\prime }}\left(x\right)-f^{{}^{\prime }}\left(x\right)g\left(x\right)+c$