Question

If $\int uv\text{'}\text{'}dx=uv\text{'}-vu\text{'}+a$, then $a=$

• A. $\int uv\text{'}\text{'}dx$
• B. $\int u\text{'}vdx$
• C. $\int uv\text{'}dx$
• D. $\int u\text{'}\text{'}vdx$

Answer 1

The correct option is D

$\int u\text{'}\text{'}vdx$

Explanation for the correct option.

Find the value of $a$.

The integration by parts is given as: $\int uv\text{'}dx=uv-\int u\text{'}vdx$.

So using that the integral $\int uv\text{'}\text{'}dx$ can be solved as:

$\begin{array}{rcl}\int uv\text{'}\text{'}dx& =& uv\text{'}-\int u\text{'}v\text{'}dx\\ & =& uv\text{'}-\left[u\text{'}v-\int u\text{'}\text{'}vdx\right]\\ & =& uv\text{'}-u\text{'}v+\int u\text{'}\text{'}vdx\end{array}$

It is given that $\int uv\text{'}\text{'}dx=uv\text{'}-vu\text{'}+a$, so the value of $a$ is $\int u\text{'}\text{'}vdx$.

Hence, the correct option is D.