Question

$\int x\left\{f\right(x^2\left)g^{\prime \prime }\right(x^2)-f^{\prime \prime }(x^2\left)g\right(x^2\left)\right\}dx$

• A. $f\left(x^2\right)g^{{}^{\prime }}\left(x^2\right)-g\left(x^2\right)f^{{}^{\prime }}\left(x^2\right)+c$
• B. $\frac{1}{2}\left\{f\right(x^2\left)g\right(x^2\left)f^{{}^{\prime }}\right(x^2\left)\right\}+c$
• C. $\frac{1}{2}\left\{f\right(x^2\left)g^{{}^{\prime }}\right(x^2)-g(x^2\left)f^{{}^{\prime }}\right(x^2\left)\right\}+c$
• D. None of these

Answer 1

The correct option is C $\frac{1}{2}\left\{f\right(x^2\left)g^{{}^{\prime }}\right(x^2)-g(x^2\left)f^{{}^{\prime }}\right(x^2\left)\right\}+c$

Put $x^2=t\Rightarrow I=\frac{1}{2}\int \left\{f\right(t\left)g^{\prime \prime }\right(t)-g(t\left)f^{\prime \prime }\right(t\left)\right\}dt=\frac{1}{2}\left\{f\right(t\left)g^{{}^{\prime }}\right(t)-\int f^{{}^{\prime }}(t\left)g^{{}^{\prime }}\right(t)-g(t\left)f^{{}^{\prime }}\right(t)+\int g^{{}^{\prime }}(t\left)f^{{}^{\prime }}\right(t\left)dt\right\}+c\therefore I=\frac{1}{2}\left\{f\right(t\left)g^{{}^{\prime }}\right(t)-g(t\left)f^{{}^{\prime }}\right(t\left)\right\}+c=\frac{1}{2}\left\{f\right(x^2\left)g^{{}^{\prime }}\right(x^2)-g(x^2\left)f^{{}^{\prime }}\right(x^2\left)\right\}+c$