Integration by parts is a method to find integrals of products.

$$\begin{align*} \text{Product Rule} &\rightarrow \text{Integration by Parts} \\ \\ \frac{d}{dx}[f(x)g(x)] &= f^\prime(x)g(x) + f(x)g^\prime(x) \\ \\ f(x)g(x) &= \int f^\prime(x)g(x) \cdot dx + \int f(x)g^\prime(x) \cdot dx \\ \\ f(x)g(x) - \int f^\prime(x)g(x) \cdot dx &= \int f(x)g^\prime(x) \cdot dx \\ \\ &\Downarrow \\ \\ \int f(x)g^\prime(x) \cdot dx &= f(x)g(x) - \int f^\prime(x)g(x) \cdot dx \end{align*}$$

$$\begin{align*} \int f(x)g^\prime(x)dx &= f(x)g(x) - \int f^\prime(x)g(x) \cdot dx \\ \\ \int x \cos{x} \cdot dx &= x \sin{x} - \int 1 \sin{x} \cdot dx \\ \\ &= x \sin{x} - (-\cos{x} + C) \\ \\ &= x \sin{x} + \cos{x} - C \\ \\ &= x \sin{x} + \cos{x} + C \end{align*}$$

$$\begin{align*} & \int f(x)g^\prime(x) \cdot dx = f(x)g(x) - \int f^\prime(x)g(x) \cdot dx \\ \\ \int (\ln{x}) dx = \int (\ln{x}) \cdot 1 dx &= \ln{x} \cdot x - \int \frac{1}{x} \cdot x dx \\ \\ &= \ln{x} \cdot x - x + C \end{align*}$$

$$\begin{align*} \int f(x)g^\prime(x) \cdot dx &= f(x)g(x) - \int f^\prime(x)g(x) \cdot dx \\ \\ \\ \int x^2 e^x dx &= x^2 e^x - \int 2x e^x dx \\ \\ \int x^2 e^x dx &= x^2 e^x - 2\underline{\int x e^x dx} \\ \\ \\ \underline{\int x e^x dx} &= x e^x - \int 1 e^x dx \\ \\ &= x e^x - e^x \\ \\ \\ \int x^2 e^x dx &= x^2 e^x - 2(\underline{x e^x - e^x}) \\ \\ &= x^2 e^x - 2x e^x + 2e^x \end{align*}$$

$$\begin{align*} \int f(x)g^\prime(x) dx &= f(x)g(x) - \int f^\prime(x)g(x) dx \\ \\ \\ \int e^x \cos{x} dx &= e^x \sin{x} - \underline{\int e^x \sin{x} dx} \\ \\ \\ \underline{\int e^x \sin{x} dx} &= e^x (-\cos{x}) - \int e^x (-\cos{x}) dx \\ \\ &= - e^x \cos{x} + \int e^x \cos{x} dx \\ \\ \\ \int e^x \cos{x} dx &= e^x \sin{x} - (\underline{- e^x \cos{x} + \int e^x \cos{x} dx}) \\ \\ &= e^x \sin{x} + e^x \cos{x} - \int e^x \cos{x} dx \\ \\ 2 \int e^x \cos{x} dx &= e^x \sin{x} + e^x \cos{x} \\ \\ \int e^x \cos{x} dx &= \frac{e^x \sin{x} + e^x \cos{x}}{2} + C \end{align*}$$

Integration by parts is a method to find integrals of products.

$$\begin{align*} \text{Product Rule} &\rightarrow \text{Integration by Parts} \\ \\ \frac{d}{dx}[f(x)g(x)] &= f^\prime(x)g(x) + f(x)g^\prime(x) \\ \\ f(x)g(x) &= \int f^\prime(x)g(x) \cdot dx + \int f(x)g^\prime(x) \cdot dx \\ \\ f(x)g(x) - \int f^\prime(x)g(x) \cdot dx &= \int f(x)g^\prime(x) \cdot dx \\ \\ &\Downarrow \\ \\ \int f(x)g^\prime(x) \cdot dx &= f(x)g(x) - \int f^\prime(x)g(x) \cdot dx \end{align*}$$

$$\begin{align*} \int f(x)g^\prime(x)dx &= f(x)g(x) - \int f^\prime(x)g(x) \cdot dx \\ \\ \int x \cos{x} \cdot dx &= x \sin{x} - \int 1 \sin{x} \cdot dx \\ \\ &= x \sin{x} - (-\cos{x} + C) \\ \\ &= x \sin{x} + \cos{x} - C \\ \\ &= x \sin{x} + \cos{x} + C \end{align*}$$

$$\begin{align*} & \int f(x)g^\prime(x) \cdot dx = f(x)g(x) - \int f^\prime(x)g(x) \cdot dx \\ \\ \int (\ln{x}) dx = \int (\ln{x}) \cdot 1 dx &= \ln{x} \cdot x - \int \frac{1}{x} \cdot x dx \\ \\ &= \ln{x} \cdot x - x + C \end{align*}$$

$$\begin{align*} \int f(x)g^\prime(x) \cdot dx &= f(x)g(x) - \int f^\prime(x)g(x) \cdot dx \\ \\ \\ \int x^2 e^x dx &= x^2 e^x - \int 2x e^x dx \\ \\ \int x^2 e^x dx &= x^2 e^x - 2\underline{\int x e^x dx} \\ \\ \\ \underline{\int x e^x dx} &= x e^x - \int 1 e^x dx \\ \\ &= x e^x - e^x \\ \\ \\ \int x^2 e^x dx &= x^2 e^x - 2(\underline{x e^x - e^x}) \\ \\ &= x^2 e^x - 2x e^x + 2e^x \end{align*}$$

$$\begin{align*} \int f(x)g^\prime(x) dx &= f(x)g(x) - \int f^\prime(x)g(x) dx \\ \\ \\ \int e^x \cos{x} dx &= e^x \sin{x} - \underline{\int e^x \sin{x} dx} \\ \\ \\ \underline{\int e^x \sin{x} dx} &= e^x (-\cos{x}) - \int e^x (-\cos{x}) dx \\ \\ &= - e^x \cos{x} + \int e^x \cos{x} dx \\ \\ \\ \int e^x \cos{x} dx &= e^x \sin{x} - (\underline{- e^x \cos{x} + \int e^x \cos{x} dx}) \\ \\ &= e^x \sin{x} + e^x \cos{x} - \int e^x \cos{x} dx \\ \\ 2 \int e^x \cos{x} dx &= e^x \sin{x} + e^x \cos{x} \\ \\ \int e^x \cos{x} dx &= \frac{e^x \sin{x} + e^x \cos{x}}{2} + C \end{align*}$$

The priciple of Integration by parts is: whose derivative $f^\prime$ simpler, who's gonna be $f(x)$