7–64. Integration review Evaluate the following integrals.
10. ∫ e^(3 - 4x) dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves an exponential function, e^(3 - 4x). To simplify, identify the inner function (3 - 4x) and its derivative (-4). This suggests using substitution.
2. Step 2: Perform substitution. Let u = 3 - 4x, which implies that du/dx = -4 or equivalently, dx = -du/4.
3. Step 3: Rewrite the integral in terms of u. Substituting u and dx, the integral becomes ∫ e^u * (-du/4). Factor out the constant -1/4 to simplify: (-1/4) ∫ e^u du.
4. Step 4: Integrate e^u with respect to u. The integral of e^u is simply e^u, so the result becomes (-1/4) * e^u + C, where C is the constant of integration.
5. Step 5: Substitute back u = 3 - 4x to return to the original variable. The final expression is (-1/4) * e^(3 - 4x) + C.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is the process of determining the area under a curve represented by a function over a specified interval. Understanding integration is crucial for evaluating integrals, as it allows us to reverse the process of differentiation.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where 'e' is the base of natural logarithms, approximately equal to 2.71828. These functions are characterized by their rapid growth or decay and are commonly encountered in calculus, especially in integration problems involving e raised to a power. Recognizing the structure of exponential functions is essential for applying integration techniques.

Substitution Method

The substitution method is a technique used in integration to simplify the process of finding an integral. It involves substituting a part of the integrand with a new variable to make the integral easier to evaluate. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative, allowing for a more straightforward integration process.