91Ó°ÊÓ

Integration by parts is a technique that helps in solving more complex integrals. The fundamental idea is to transform the integral of a product of functions into an easier form. The formula is given by: \ \ \ \ \[ \int u \, dv = uv - \int v \, du \] \ \ In this formula: \ \ \ \ \begin{itemize} \ \ \item \ (u) is a function that we choose to differentiate. \ \ \item \ (dv) is the part we integrate. \ \ \ \item \ Results in (du) as the derivative of u. \ \ \item \ (v) is the result of integrating dv. \ \ \ \ \end{itemize} \ \ \ Let's take a closer look at our example. We start with the definite integral: \ \ \ \ \[ \int_{-1}^{1} 2x e^x \, dx \] \ \ \ \ Here, we'll let: \ \ \ \ \[ \ u = 2x \] \ \ \and: \ \ \ \ \ \ \ dv = e^x \, dx \ \ \ \ Now, differentiate u and integrate dv: \ \ \ \ \ du = 2 \, dx \ \ \ \ \ v = e^x \ \ \ \ Substitute these into the formula: \ \ \ \ \ \ \[ \int_{-1}^{1} 2x e^x \, dx = \left[ 2x e^x \right]_{-1}^{1} - \int_{-1}^{1} 2e^x \, dx \right. \] \ \ \ This breaks the original problem down into manageable pieces. \ \ \

A definite integral computes the area under a curve between two specific points. This means it gives a numerical value rather than a general function. Consider the definite integral: \ \ \ \ \[ \int_{a}^{b} f(x) \, dx \] \ \ \ \ \ where \ \ a \ \ and \ \ b \ \ are the limits of integration. \ \ In our example: \ \ a = -1 \ \ and \ \ b = 1. \ \ Substituting these limits into the solution, we evaluate the boundary terms: \ \ \ \ \[ \left[ 2x e^x \right]_{-1}^{1} = \left(2(1)e^1 - 2(-1)e^{-1}\right) = 2e + \frac{-2}{e} \] \ \ \ This evaluates the function at the upper (1) and lower (-1) limits and combines the results. \ \

The exponential function, denoted as \ \ e^x, \ \ is fundamental in mathematics. This function grows and decays rapidly and appears frequently in various scientific disciplines. Its key properties include: \ \ \ \ \ \begin{itemize} \ \ \ \item \ \ The derivative of \ \ e^x \ \ is \ \ e^x. \ \ \ \item \ \ \ The integral of \ \ e^x \ \ is \ \ e^x + C. \ \ \ \item \ \ \ The function is positive for all real numbers. \ \ \ \item \ \ \ It crosses the y-axis at \ \ (0,1). \ \ \ \end{itemize} \ \ \ In our example, integrating \ \ 2e^x \ \ involves the step: \ \ \ \ \ \[ \ \ 2e^x \, dx = 2(e^x) \right]_{-1}^{1} = 2(e^1 - e^{-1}) = 2e - \frac{2}{e} \ \right\text. \] Combining this with our previous result, we achieve the final answer: \ \[ \ (2e + \frac{-2}{e}) - (2e - \frac{2}{e}) = \frac{4}{e} \] which evaluates the integral fully. Understanding exponential functions is key to working through these types of integrals smoothly. \