Chapter 3, Problem 408RE

To determine

To check: Whether the given statement is true or false and also justified it.

Answer to Problem 408RE

The given statement is false because ${\displaystyle \int e^x\sin x}dx$ can be integrated by parts.

Explanation of Solution

Given information:

The given statement is, “ ${\displaystyle \int e^x\sin x}dx$ cannot be integrated by parts”.

Concept used:

Integration by parts,

  ${\displaystyle \int uv}dx=u{\displaystyle \int v}dx-{\displaystyle \int \left(u^{\prime }{\displaystyle \int v}dx\right)}dx$ .

Calculation:

Consider ${\displaystyle \int e^x\sin x}dx$

Let $u=e^x\Rightarrow du=e^{x}dx$

And $dv=\sin xdx$

  $\Rightarrow {\displaystyle \int dv}={\displaystyle \int \sin xdx\Rightarrow v=-\cos x}$

So, using integration by parts, we get

  $\begin{array}{c}{\displaystyle \int e^x\sin x}dx=e^x(-\cos x)-{\displaystyle \int (-\cos x)e^{x}dx}\\ =-e^x\cos x+{\displaystyle \int e^x\cos xdx}+c_1\end{array}$

  $\begin{array}{l}\text{Let }u=e^x\Rightarrow du=e^{x}dx\\ \text{and }dv=\cos xdx\Rightarrow {\displaystyle \int dv}={\displaystyle \int \cos xdx\Rightarrow v=\sin x}\end{array}$

Therefore, using again integration by parts, we get

  ${\displaystyle \int e^x\sin x}dx=-e^x\cos x+e^x\sin x-{\displaystyle \int e^x\sin x}dx+c_2$

After adding ${\displaystyle \int e^x\sin x}dx$ on both sides, we get

  $\text{2}{\displaystyle \int e^x\sin x}dx=-e^x\cos x+e^x\sin x+c_2$

Thus, ${\displaystyle \int e^x\sin xdx=\frac{1}{2}{e}^x\left(\sin x-\cos x\right)+C}$

Here, C is constant of integration.

Conclusion:

Hence, ${\displaystyle \int e^x\sin x}dx$ can be integrated by parts.