Question

Fill in the blanks to complete each of the following theorem statements:

If $u$and $v$are functions such that $u\text{'}\left(x\right)v\left(x\right)+u\left(x\right)v\text{'}\left(x\right)$is _____, then we can use the product rule in reverse to obtain$\int u\text{'}\left(x\right)v\left(x\right)+u\left(x\right)v\text{'}\left(x\right)dx=\_\_\_\_\_$.

Short answer

If $u$and $v$are functions such that $u\text{'}\left(x\right)v\left(x\right)+u\left(x\right)v\text{'}\left(x\right)dx$is integrable, then we can use the product rule in reverse to obtain$\int u\text{'}\left(x\right)v\left(x\right)+u\left(x\right)v\text{'}\left(x\right)dx=u\left(x\right)v\left(x\right)+C$.

Step-by-step solution

Step 1. Given information 

If $u$and $v$are functions such that $u\text{'}\left(x\right)v\left(x\right)+u\left(x\right)v\text{'}\left(x\right)dx$is _____, then we can use the product rule in reverse to obtain $\int u\text{'}\left(x\right)v\left(x\right)+u\left(x\right)v\text{'}\left(x\right)dx=\_\_\_\_\_$.

Step 2. Filling in the blanks to complete the theorem statements 

If $u$and $v$are functions such that $u\text{'}\left(x\right)v\left(x\right)+u\left(x\right)v\text{'}\left(x\right)dx$ is integrable, then we can use the product rule in reverse to obtain $\int u\text{'}\left(x\right)v\left(x\right)+u\left(x\right)v\text{'}\left(x\right)dx=u\left(x\right)v\left(x\right)+C$.

By using reverse of product rule, we get the integral above.