Solution assignment 02 Integration by parts

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Assignment 2

Calculate:

$\displaystyle\int{e^x\sin(x)}dx$

Solution

We can calculate this integral by applying integration by parts. We do this in two steps. We can write the integral as:

$\displaystyle\int{e^x\sin(x)}dx=-\displaystyle\int{e^x}dcos(x)=$

$=-e^x\cos(x)+\displaystyle\int{\cos(x)}de^x=$

$=-e^x\cos(x)+\displaystyle\int{e^x\cos(x)}dx$

It seems that this approach has not helped us a lot. The original integrand is the product $e^x\sin(x)$ and the result is the integrand $e^x\cos(x)$ .
However we write:

$\displaystyle\int{e^x\cos(x)}dx=\displaystyle\int{e^x}d\sin(x)=$

$=e^x\sin(x)-\displaystyle\int{\sin(x)}de^x=$

$=e^x\sin(x)-\displaystyle\int{e^x\sin(x)}dx$

We take:

$I=\displaystyle\int{e^x\sin(x)}dx$

and get:

$I=-e^x\cos(x)+e^x\sin(x)-I$

We solve this equation in $I$ :

$\displaystyle\int{e^x\sin(x)}dx=\displaystyle\frac{1}{2}e^x[\sin(x)-\cos(x)]+C$

Verify the result by differentiation.

Return to Assignments Integration by parts

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Solution assignment 02 Integration by parts

Assignment 2

Solution

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