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.