Complete The Table By Identifying And Calculating The Integr

Question

Complete The Table By Identifyinguanddufor The Integralfgxgxdxu Complete the table by identifying u and du for the integral. f g ( x ) g ' ( x ) dx u = g ( x ) du = g ' ( x ) dx (tan( x )) 5 (sec( x )) 2 dx u = du = dx Find the indefinite integral and check the result by differentiation. (Use C for the constant of integration.)

Dr. Jack HW Helper · Accepted Answer

The task involves solving the integral of the function f(x) = tan(x) and verifying the solution through differentiation. The process begins with selecting an appropriate substitution to simplify the integral. To systematically approach this problem, we start by creating a table to identify the parts of the integral, specifically the functions u and du, which allows us to apply the method of integration by parts effectively. Given the integral ∫tan(x) dx, recall that tangent can be expressed as a ratio of sine and cosine functions: tan(x) = sin(x)/cos(x). This inherently suggests a substitution approach, often involving the reciprocal or derivatives of these trigonometric functions. Alternatively, recognizing that tan(x) is the derivative of -ln|cos(x)| can lead to a straightforward integration without direct substitution. Nevertheless, for the purpose of practicing integration techniques, we can employ substitution to verify their effectiveness. Step 1: Identify u and du Considering the integral ∫tan(x) dx, we observe that the derivative of -ln|cos(x)| is tan(x), which suggests choosing u = cos(x). Differentiating, we find du = -sin(x) dx. However, since the goal is to find an integral that simplifies when substituting u, an alternative, more straightforward approach is to use the substitution u = sin(x), because du = cos(x) dx, which can simplify the original integral. Step 2: Setting up the substitution Let’s set u = sin(x). Then, du = cos(x) dx. Remember, the integral involves tan(x) = sin(x)/cos(x). Rewriting the integral: ∫tan(x) dx = ∫(sin(x)/cos(x)) dx = ∫(u / sqrt(1 - u^2)) du, assuming u = sin(x); however, this adds complexity. Alternatively, a more straightforward substitution is to recognize that the integral of tan(x) dx directly results in -ln|cos(x)| + C, because the derivative of -ln|cos(x)| is tan(x). Step 3: Direct integration as an alternative Since the integral of tan(x) dx is well-known, it evaluates to: ∫tan(x) dx = -ln|cos(x)| + C. To verify, d

Complete The Table By Identifying And Calculating The Integr

Complete The Table By Identifyinguanddufor The Integralfgxgxdxu

Paper For Above instruction

Step 1: Identify u and du

Step 2: Setting up the substitution

Step 3: Direct integration as an alternative

Conclusion and final answer

References