Question

If $\int f\left(x\right)dx=g\left(x\right)$, then $\int f\left(x\right)g\left(x\right)dx=$

• A. $\frac{1}{2}{f}^2\left(x\right)$
• B. $\frac{1}{2}{g}^2\left(x\right)$
• C. $\frac{1}{2}\left[g^{\prime }\right(x){]}^2$
• D. $f^{\prime }\left(x\right)g\left(x\right)$

Answer 1

Answer: (B)

$\frac{1}{2}{g}^2\left(x\right)$

$\int f\left(x\right)dx=g\left(x\right)$

So, $f\left(x\right)=g^{\prime }\left(x\right)$

We have,

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

$\int f\left(x\right)g\left(x\right).dx=g^2\left(x\right)-\int f\left(x\right)g\left(x\right).dx$

$\int f\left(x\right)g\left(x\right).dx=\frac{g^2\left(x\right)}{2}$

(b)