Question

Evaluate the following integrals. Include absolute values only when needed. $$\int 7^{2 x} d x$$

Short answer

Question: Evaluate the following integral: $$\int 7^{2 x} d x$$ Answer: $$\int 7^{2 x} d x = \frac{1}{2}( \frac{7^{2x}}{\ln 7} + C)$$

Step-by-step solution

Substitution for integration

Let's make the substitution \(u = 2x\). Then, we find the differential \(d u\): $$d u = 2 \, d x$$ Now, we rearrange the equation for \(dx\). Thus, $$dx = \frac{1}{2} du$$ Now, we rewrite the integral in terms of \(u\): $$\int 7^{2 x} d x = \int 7^{u}\frac{1}{2} d u$$

Integration

Now, we can integrate this function with respect to \(u\) $$\int 7^{u}\frac{1}{2} d u = \frac{1}{2} \int 7^{u} d u$$ Using the formula for exponential functions where $$\int a^{u} du = \frac{a^u}{\ln a} + C$$ where C is the constant of integration. We'll apply this formula to our integral: $$\frac{1}{2} \int 7^{u} d u = \frac{1}{2}( \frac{7^u}{\ln 7} + C)$$

Substitute back

Now, we substitute \(u\) back in terms of \(x\): $$\frac{1}{2}( \frac{7^u}{\ln 7} + C) = \frac{1}{2}( \frac{7^{2x}}{\ln 7} + C)$$

Writing the final answer

Finally, write the solution for the given integral: $$\int 7^{2 x} d x = \frac{1}{2}( \frac{7^{2x}}{\ln 7} + C)$$

Key concepts

Integration by Substitution

When integrating complex functions, integration by substitution, commonly known as u-substitution, is a powerful technique. It involves replacing a part of the integral with a new variable, 'u', which simplifies the integral into an easier form. This method is particularly useful when dealing with functions within functions, like compound expressions involving exponentials or trigonometric functions.

For instance, in evaluating \(\int 7^{2x} dx\), you can simplify the integration process by letting \(u = 2x\). Following this substitution, the differential \(du = 2dx\) is obtained, which helps to rewrite the original integral in terms of 'u'. The equation for \(dx\) is then rearranged, giving \(dx = \frac{1}{2}du\). This substitution makes the integral more manageable and sets the stage for applying exponential integration rules.

Indefinite Integrals

An indefinite integral, represented by the Integral sign (\(\int\)), denotes the antiderivative of a function. It’s called indefinite because it represents a family of functions, rather than a specific function, due to the addition of an arbitrary constant \(C\), known as the constant of integration. Essentially, it is the reverse process of differentiation.

The calculation of an indefinite integral aims to find a function whose derivative gives the integrand. For example, if you have \(\int f(x) dx\), you're looking to find a function \(F(x)\) such that \(F'(x) = f(x)\). The indefinite integral of the function will be \(F(x) + C\), where 'f(x)' is the integrand and \(F(x)\) is the antiderivative.

Antiderivatives

Antiderivatives are the flip side of derivatives. They are the functions that, when differentiated, yield the original function from which they were derived. In the context of integration, finding an antiderivative means solving for the indefinite integral of a given function.

For the exponential function in our example \(7^{2x}\), the antiderivative can be systematically found using integration rules. It is of utmost importance to recognize that finding the antiderivative is not just about reversing differentiation but also about understanding the relationship between the functions and their rates of change.

Exponential Integration Rules

Exponential integration rules are specific guidelines for integrating functions of the form \(a^x\), where 'a' is a constant. The rule states that the integral of \(a^x\) with respect to 'x' is \(\frac{a^x}{\ln(a)} + C\), where \(\ln(a)\) is the natural logarithm of 'a', and \(C\) is the constant of integration.

In the provided exercise, after performing the substitution, the rule is applied to integrate \(7^u\) with \(u = 2x\). The resulting function, \(\frac{7^u}{\ln(7)}\), is then multiplied by \(\frac{1}{2}\) to accommodate for the earlier substitution step. Finally, 'u' is substituted back with \(2x\) to express the antiderivative in terms of the original variable, yielding the final answer \(\frac{1}{2}(\frac{7^{2x}}{\ln 7} + C)\), where the constant of integration shows that there can be many functions whose derivative gives the original function.