AP Calculus AB 6.9 Integration By "u" Substitution (Example 3 e^3x)

Please subscribe!    / nickperich   AP Calculus AB 6.9: Integration by "u" Substitution (Example 3: \( e^{3x} \)) --- #### Overview In this section, students learn how to apply the "u" substitution technique to integrals involving exponential functions. This method simplifies the integration process by changing the variable to facilitate easier calculation. --- #### Key Concepts 1. *Understanding "u" Substitution:* The "u" substitution technique is effective in transforming integrals into simpler forms, particularly when dealing with compositions of functions, such as exponential functions. The process involves choosing a substitution \( u = g(x) \) that simplifies the integrand, determining the differential \( du \), and re-expressing the integral in terms of \( u \). 2. *Example with an Exponential Function:* Consider the integral: \[ \int e^{3x} \, dx \] A suitable choice for \( u \) is \( u = 3x \). This substitution will simplify the exponent in the integrand. 3. *Finding \( du \):* Compute the differential: \[ du = 3 \, dx \quad \Rightarrow \quad dx = \frac{du}{3} \] 4. *Substituting \( u \) and \( dx \):* Replace \( dx \) in the integral: \[ \int e^{3x} \, dx = \int e^{u} \cdot \frac{du}{3} \] This can be simplified as: \[ \frac{1}{3} \int e^{u} \, du \] 5. *Integrating the New Function:* The integral of \( e^{u} \) is straightforward: \[ \int e^{u} \, du = e^{u} + C \] Thus: \[ \frac{1}{3}(e^{u} + C) = \frac{1}{3}e^{u} + C \] 6. *Substituting Back to the Original Variable:* Replace \( u \) with \( 3x \): \[ \frac{1}{3} e^{3x} + C \] --- #### Conclusion Using "u" substitution is an effective method for integrating exponential functions like \( e^{3x} \). By choosing an appropriate substitution and following the steps of substitution, students can simplify complex integrals into more manageable forms. This example illustrates how to handle integrals involving exponential functions efficiently, ultimately leading to a clean expression in terms of the original variable. 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