example of integration by parts involving algebraic manipulation

For some integrals which require integration by parts, it may be to treat the integral like a variable and solve for it. For example, consider the integral

$$\int e^x\cos xdx.$$

Using integration by parts with the substitutions

we obtain

$$\int e^x\cos xdx=e^x\cos x+\int e^x\sin xdx.$$

Using integration by parts on the integral on the right hand side with the substitutions

we obtain

$$\int e^x\cos xdx=e^x\cos x+e^x\sin x-\int e^x\cos xdx.$$

The “trick” is to add $\int e^x\cos xdx$ to both sides of the equation. Some people find this concept surprising at first sight, especially since most people who are taking calculus for the first time do not use equations when showing their work for integration. For integrals such as $\int e^x\cos xdx$, writing out an equation is essential.

After adding $\int e^x\cos xdx$ to both sides of the above equation, we will need a $+C$ on the right hand side. Thus, we obtain

$$2\int e^x\cos xdx=e^x\cos x+e^x\sin x+C.$$

Therefore, we can figure out what $\int e^x\cos xdx$ is by dividing both sides by $2$, which yields

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

On the other hand, since $C$ is an arbitrary constant, we generally write

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

with the understanding that the constant $C$ in the final equation may not have the same value as $C$ appearing in equations in previous steps.

Title	example of integration by parts involving algebraic manipulation
Canonical name	ExampleOfIntegrationByPartsInvolvingAlgebraicManipulation
Date of creation	2013-03-22 17:39:52
Last modified on	2013-03-22 17:39:52
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	10
Author	Wkbj79 (1863)
Entry type	Example
Classification	msc 97D70
Classification	msc 26A36
Related topic	ALectureOnIntegrationByParts