Question

Evaluate the following integrals. $$\int_{0}^{5} 5^{5 x} d x$$

Short answer

Answer: The value of the definite integral is \(\frac{5^{25} - 1}{5\ln5}\).

Step-by-step solution

Find the Antiderivative

To find the antiderivative of \(5^{5x}\), we will use the formula for the antiderivative of an exponential function, which is \(\frac{a^{kx}}{k \ln a}\), where \(a\) is the base, and \(k\) is the exponential rate. Let's apply this formula to our function: $$\int 5^{5x} dx = \frac{5^{5x}}{5\ln5} + C$$

Apply the Limits of Integration

Now that we have the antiderivative, let's use the fundamental theorem of calculus to evaluate the definite integral. We will subtract the value of the antiderivative at the lower limit (\(0\)) from the value of the antiderivative at the upper limit (\(5\)): $$\int_{0}^{5} 5^{5x} dx = \left[\frac{5^{5x}}{5\ln5}\right]_{0}^{5}$$ $$= \frac{5^{5(5)}}{5\ln5} - \frac{5^{5(0)}}{5\ln5}$$

Simplify the Expression

Finally, let's simplify the expression to find the value of the definite integral: $$= \frac{5^{25}}{5\ln5} - \frac{5^{0}}{5\ln5}$$ $$= \frac{5^{25} - 1}{5\ln5}$$ Thus, the value of the definite integral is: $$\int_{0}^{5} 5^{5x} dx = \frac{5^{25} - 1}{5\ln5}$$

Key concepts

Antiderivatives

Antiderivatives are the reverse process of differentiation. Basically, if you know the derivative of a function, finding its antiderivative means finding the original function before it was differentiated. In another way, if you have a family of functions whose derivative is a given function, one of them will be the antiderivative.

For exponential functions like the one in our problem, the antiderivative can be found by using a special rule. This rule involves dividing by the rate, often called "k", and the natural logarithm of the base of the exponential function.

When we are given an exponential function like \(5^{5x}\), the formula helps us to find its antiderivative as \(\frac{5^{5x}}{5 \ln 5} + C\), where \(C\) is the constant of integration. This step is crucial in problems involving definite integrals because it sets the stage for evaluating the integral over specified limits.

Exponential Functions

Exponential functions are a type of function involving constants raised to a variable power, such as \(a^x\). They are important in both mathematics and real-world applications, modeling everything from population growth to radioactive decay.

In our integral, the function \(5^{5x}\) is an exponential function where \(5\) is the base and \(5x\) is the exponent. This means as \(x\) increases, \(5^{5x}\) grows very quickly.

One important property of exponential functions is that their rate of change accelerates over time, making them different from linear functions which have a constant rate of change. When integrating exponential functions, this rapid growth must be accounted for, usually by using logarithmic expressions as seen in the antiderivative \(\frac{5^{5x}}{5 \ln 5} + C\). This reflects the adjustment for the increasing rate.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus ties together the concepts of differentiation and integration, the two main branches of calculus. It provides a powerful connection between the two by stating that differentiation and integration are inverse processes. This theorem allows us to evaluate defined integrals by making use of the antiderivative.

In the context of definite integrals, the limitation is accounted for by evaluating the antiderivative at specific bounds. For our exercise, the limits of integration were from \(0\) to \(5\).

After finding the antiderivative \(\frac{5^{5x}}{5 \ln 5} + C\), we evaluate it at the upper limit \(5\) and the lower limit \(0\), then subtract the two results. This step transforms an indefinite integral into a specified value, making calculus a powerful tool for solving practical problems.