Problem 35 - Calculus: Early Transcendentals

Question

Evaluate the integral. \( \displaystyle \int^{1}_{0} (x^{10} + 10^x)\,dx \)

Step-by-Step Solution

Answer

\( \frac{1}{11} + \frac{9}{\ln 10} \)

Step 1: Split the Integral

First, note that the integral contains two terms: \(x^{10}\) and \(10^x\). Therefore, we can split the given integral into two separate integrals: \[ \displaystyle \int_{0}^{1} (x^{10} + 10^x)\,dx = \displaystyle \int_{0}^{1} x^{10}\,dx + \displaystyle \int_{0}^{1} 10^x\,dx. \]

Step 2: Evaluate the First Integral

Now, let's evaluate the integral \(\displaystyle \int_{0}^{1} x^{10}\,dx\). The antiderivative of \(x^{10}\) is \(\frac{x^{11}}{11}\). Using the Fundamental Theorem of Calculus, we find:\[ \left[ \frac{x^{11}}{11} \right]_{0}^{1} = \frac{1^{11}}{11} - \frac{0^{11}}{11} = \frac{1}{11}. \]

Step 3: Evaluate the Second Integral

Next, we evaluate the integral \(\displaystyle \int_{0}^{1} 10^x\,dx\). The antiderivative of \(10^x\) is \(\frac{10^x}{\ln 10}\). Again, using the Fundamental Theorem of Calculus, we find:\[ \left[ \frac{10^x}{\ln 10} \right]_{0}^{1} = \frac{10^1}{\ln 10} - \frac{10^0}{\ln 10} = \frac{10}{\ln 10} - \frac{1}{\ln 10} = \frac{9}{\ln 10}. \]

Step 4: Combine the Results

Finally, add the results of the two integrals to get the value of the original integral:\[ \int_{0}^{1} (x^{10} + 10^x)\,dx = \frac{1}{11} + \frac{9}{\ln 10}. \]

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a powerful principle that links the concept of differentiation with integration, two core processes in calculus. To put it simply:

• The first part of the theorem enables us to find a function's integral by evaluating its antiderivative at the boundaries of a given interval.
• The second part tells us how to differentiate an integral function with respect to its upper limit.

The theorem's magic lies in its ability to turn the complex process of finding the total area under a curve into something straightforward. Consider the example in the solution: we calculated the integral of the polynomial function using its antiderivative. We applied the theorem by evaluating this antiderivative from 0 to 1. This step converts the continuous sum of areas (or integration) into a simple subtraction. This is why the Fundamental Theorem of Calculus is such a cornerstone in our understanding of calculus. Whenever faced with an indefinite integral, you can thank this theorem for making it manageable through its relationship with antiderivatives. By grasping this fundamental connection, complex integrals become approachable, which is why it's crucial for mastering calculus.

Polynomial Integration

Polynomial Integration involves finding the integral of polynomial expressions. It is one of the simplest forms of integration due to its straightforward nature. To integrate a polynomial like \(x^{10}\), follow these steps:

• Increase the power of the variable by one.
• Divide the result by the new power.
• Add the constant of integration \(C\) if it's an indefinite integral.

For the example of \(x^{10}\), its antiderivative becomes \(\frac{x^{11}}{11}\). This is because, according to the rule, you add one to the exponent (10 becomes 11) and divide by the new exponent (11).In our solution, we integrated \(x^{10}\) from 0 to 1 using definite integration, which means evaluating this antiderivative at the interval boundaries and subtracting the results.Polynomial integration is fundamental because many more complex integrals can be simplified to polynomials, making it an essential skill for calculus students.Mastering polynomial integration builds a strong foundation for tackling more complicated expressions involving higher-level calculus techniques.

Exponential Function Integration

Integrating exponential functions like \(10^x\) may seem daunting at first, but they're straightforward with practice. When you integrate \(a^x\) where \(a\) is a constant, remember the following steps:

• The antiderivative of \(a^x\) is \(\frac{a^x}{\ln(a)}\).
• This formula results from the chain rule and the properties of logarithms.
• Always evaluate using the bounds if it's a definite integral, subtracting the lower bound evaluation from the upper.

In the provided example, \(10^x\) was integrated over the interval from 0 to 1. We used its antiderivative \(\frac{10^x}{\ln(10)}\), calculating this at both bounds. We applied the Fundamental Theorem of Calculus by evaluating this expression at 1 and subtracting the evaluation at 0.Understanding how to integrate the exponential function is key to many applications in science and engineering, like growth models and decay processes.By mastering these fundamental concepts, you'll gain a deeper respect for how exponential growth and decay are represented in calculus, making integration tasks simpler.