Chapter 7, Problem 1CRQ

To determine

To fill: “The integration by parts formula is obtained by reversing the _______ Rule. The formula for integration by parts is ${\displaystyle \int udv=}$_______. In choosing u and dv, we want du to be simpler than _______ and dv to be _____ _____ ______”.

Answer to Problem 1CRQ

The integration by parts formula is obtained by reversing the product Rule. The formula for integration by parts is ${\displaystyle \int udv=}uv-{\displaystyle \int vdu}$. In choosing u and dv, we want du to be simpler than u and dv to be easy to integrate”.

Explanation of Solution

Choose u and v as differentiable functions with respect to x.

Then, by the product rule, $\frac{d}{dx}\left(uv\right)=uv\text{'}+vu\text{'}$.

Integrate and simplify this equation as follows.

$\begin{array}{c}{\displaystyle \int \frac{d}{dx}\left(uv\right)}={\displaystyle \int \left(uv\text{'}+vu\text{'}\right)}\\ uv={\displaystyle \int uv\text{'}}+{\displaystyle \int vu\text{'}}\\ {\displaystyle \int uv\text{'}}=uv-{\displaystyle \int vu\text{'}}\end{array}$

So, the integration by parts formula is obtained by reversing the product rule.

The formula for integration by parts is ${\displaystyle \int udv=}uv-{\displaystyle \int vdu}$.

This method y depends on the choice of u and dv.

There are some guidelines to choose u and dv such that, dv should be simpler than u.

In this formula, dv is the part to integrate. So, dv should be easy to integrate.

Thus, “the integration by parts formula is obtained by reversing the product Rule. The formula for integration by parts is ${\displaystyle \int udv=}uv-{\displaystyle \int vdu}$. In choosing u and dv, we want du to be simpler than u and dv to be easy to integrate”.