Find the derivative of the function given by f ( x ) = (1 + x ) (1 + x 2) (1 + x 4) (1 + x 8) and hence find fโฒ (1). If u, v and w are functions of x , then show that ๐ ๐ ๐ฅ (๐ข ๐ฃ ๐ค) = ๐ ๐ข ๐ ๐ฅ ๐ฃ ๐ค + ๐ข ๐ ๐ฃ ๐ ๐ฅ ๐ค + ๐ข ๐ฃ ๐ ๐ค ๐ ๐ฅ in two ways-first by repeated application of product ... Evaluate the integral of log x divided by x to find the solution for this mathematical problem. Mathematically, we can write the formula for the integration of log x , โซ log x dx = xlogx - x + C (OR) โซln x dx = xlnx - x + C, where log x or ln x are the natural logarithmic function. Further in this article, we will evaluate the integral of ln x or log x with base e using the integration by parts formula. To evaluate the integral โซ logxdx, we can use integration by parts. Let's go through the solution step by step. Step 1: Set up the integral Let I = โซ logxdx. Step 2: Choose u and dv We will use integration by parts, which states: โซ udv= uvโโซ vdu Here, we choose: - u= logx (which we will differentiate) - dv= dx (which we will integrate ) Step 3: Differentiate u and integrate dv Now we need to find du and v: - Differentiate u: du = 1 x dx - Integrate dv: v= x Step 4: Apply integration ...
