AP Calculus AB 6.9 Integration By "u" Substitution (Example 7 Cosine and LN)

Please subscribe!    / nickperich   AP Calculus AB 6.9: Integration by "u" Substitution (Example 7: \(\cos(x) \ln(x)\)) --- #### Overview In this example, students will learn how to apply the "u" substitution technique to integrate the function \(\cos(x) \ln(x)\). This example highlights the usefulness of substitution when dealing with products of trigonometric and logarithmic functions. --- #### Key Concepts 1. *Understanding "u" Substitution:* The "u" substitution technique simplifies the integration process for functions that are products of different types of functions, making it easier to find the antiderivative. 2. *The Integral to Solve:* Consider the integral: \[ \int \cos(x) \ln(x) \, dx \] 3. *Choosing the Substitution:* We choose: \[ u = \ln(x) \quad \Rightarrow \quad du = \frac{1}{x} \, dx \quad \Rightarrow \quad dx = x \, du \] From the substitution, we can express \(x\) in terms of \(u\): \[ x = e^u \] 4. *Substituting in Terms of \(u\):* The integral can be rewritten as: \[ \int \cos(x) \ln(x) \, dx = \int \cos(e^u) \cdot u \cdot e^u \, du \] Now we need to express \(\cos(x)\) in terms of \(u\), which is not straightforward. Instead, we should revert back to a different method or an alternative substitution. 5. *Re-evaluating the Approach:* Since \(\cos(x)\) cannot be simplified effectively through direct substitution, we can utilize integration by parts, where: Let \(v = \ln(x)\) and \(dw = \cos(x) \, dx\). Then, differentiate and integrate accordingly: \(dv = \frac{1}{x} \, dx\) \(w = \sin(x)\) 6. *Applying Integration by Parts:* Using the integration by parts formula: \[ \int v \, dw = vw - \int w \, dv \] Substitute the values: \[ \int \ln(x) \cos(x) \, dx = \ln(x) \sin(x) - \int \sin(x) \cdot \frac{1}{x} \, dx \] 7. *Simplifying Further:* The integral \(\int \sin(x) \cdot \frac{1}{x} \, dx\) does not have a simple antiderivative in terms of elementary functions, and it may be expressed in terms of special functions or solved numerically. 8. *Final Result:* Thus, the solution to the integral \(\int \cos(x) \ln(x) \, dx\) is expressed as: \[ \int \cos(x) \ln(x) \, dx = \ln(x) \sin(x) - \int \frac{\sin(x)}{x} \, dx + C \] This highlights how integration by parts can be effectively utilized when simple "u" substitution is insufficient. --- #### Conclusion In this example, the integral of \(\cos(x) \ln(x)\) demonstrates the process of integration by parts rather than relying solely on "u" substitution. The key takeaway is that while "u" substitution is a powerful tool, some integrals may require a combination of techniques to reach the solution. The final expression captures the result in terms of logarithmic and trigonometric functions, indicating the complexity of the integral. I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: / nickperich Nick Perich Norristown Area High School Norristown Area School District Norristown, Pa #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study