Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals
∫(e³ˣ ⁺¹ d𝓍

Verified step by step guidance

1. Recognize that the integral involves an exponential function with a linear term in the exponent. To simplify, use the substitution $u\; =\; 3𝓍\; +\; 1$, where $a\; =\; 3$ and $b\; =\; 1$.
2. Differentiate the substitution $u\; =\; 3𝓍\; +\; 1$ with respect to $𝓍$ to find $du/d𝓍\; =\; 3$, or equivalently, $du\; =\; 3\; d𝓍$.
3. Solve for $d𝓍$ in terms of $du$: $d𝓍\; =\; du/3$. Substitute this and $u$ into the integral, replacing $3𝓍\; +\; 1$ with $u$ and $d𝓍$ with $du/3$.
4. The integral now becomes $\int e^u\; (du/3)$. Factor out the constant $1/3$ to simplify the expression to $(1/3)\; \int e^u\; du$.
5. Evaluate the integral of $e^u$, which is simply $e^u$. Replace $u$ with the original substitution $3𝓍\; +\; 1$ to return to the variable $𝓍$. The final result will include a constant of integration, $C$.

Key Concepts

Substitution Method in Integration

The substitution method is a technique used in calculus to simplify the process of integration. By substituting a part of the integrand with a new variable, it transforms the integral into a more manageable form. This method is particularly useful when dealing with composite functions, allowing for easier integration by changing the variable of integration.

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits of integration and include a constant of integration, typically denoted as 'C'. Understanding indefinite integrals is crucial for finding antiderivatives and solving differential equations, as they provide the general solution to integration problems.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = e^(kx), where 'e' is the base of natural logarithms and 'k' is a constant. These functions are characterized by their rapid growth and unique properties, such as the fact that their derivative is proportional to the function itself. Recognizing the behavior of exponential functions is essential for evaluating integrals involving them, particularly in the context of substitution.