Problem 52 - Calculus

Question

Use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi} x^{4} \cos \left(2 x^{5}\right) d x $$

Step-by-Step Solution

Answer

The definite integral evaluates to 0.

Step 1: Choose a substitution

To apply the substitution rule, we start by looking for a substitution that simplifies the integrand. Notice that the argument of the cosine function, \(2x^5\), suggests substitution. Set \(u = 2x^5\).

Step 2: Compute the derivative

Differentiate the substitution equation \(u = 2x^5\) to find \(du/dx\). We have \(du = 10x^4 dx\). Rewriting, we get \(dx = \frac{du}{10x^4}\).

Step 3: Substitute into the integral

Substitute \(u = 2x^5\) and \(dx = \frac{du}{10x^4}\) into the original integral. The integral becomes \(\int_{0}^{\pi} x^4 \cos(u) \frac{du}{10x^4}\). Simplifying, we get \(\int \frac{\cos(u)}{10} du\).

Step 4: Adjust limits of integration

Since we changed variables from \(x\) to \(u\), we need to adjust the limits of integration. When \(x = 0\), \(u = 2 \cdot 0^5 = 0\); when \(x = \pi\), \(u = 2 \cdot (\pi)^5\). So, the new limits for \(u\) are 0 to \(2\pi^5\).

Step 5: Integrate with respect to \(u\)

The integral is now \(\int_{0}^{2\pi^5} \frac{\cos(u)}{10} du\). The antiderivative of \(\cos(u)\) is \(\sin(u)\). Therefore, this becomes \(\frac{1}{10} \left[ \sin(u) \right]_{0}^{2\pi^5}\).

Step 6: Evaluate the definite integral

Evaluate the integral from the adjusted limits: \(\frac{1}{10} \left( \sin(2\pi^5) - \sin(0) \right)\). Since \(\sin(2\pi^5)\) and \(\sin(0)\) both equal 0, the result is 0.

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus, representing the net area under a curve over a given interval. When you integrate a function from point \(a\) to point \(b\), this is what you are doing.

• This integration has endpoints, called limits of integration, which are important as they specify the range over which the area is summed.
• In this particular problem, the limits are from \(0\) to \(\pi\), which means we are finding the net area from \(x = 0\) to \(x = \pi\).
• Definite integrals can be solved in various ways, but when the integration is complicated as it involves complex functions like trigonometric functions, techniques such as substitution are employed.

Substitution helps simplify the integral, allowing us to evaluate it more easily.

Change of Variables

The 'Change of Variables' or substitution method is a powerful tool used in integration to simplify complex integrals. It involves substituting part of the integral with a new variable which can make the process of finding the integral much easier.

• In our example, the substitution \(u = 2x^5\) was chosen based on the argument of the cosine function \(\cos(2x^5)\).
• This simplifies the expression as it reduces the dependency on \(x\) inside the integral, making integration straightforward.
• After substitution, you need to adjust the limits of integration to reflect the new variable, \(u\). The endpoints \(0\) and \(\pi\) become \(0\) and \(2\pi^5\), respectively.

This adjustment is crucial, as it ensures that the integral remains the same but is expressed more simply in terms of \(u\).

Trigonometric Functions

Trigonometric functions such as \(\sin, \cos,\) and \(\tan\) play a significant role in integration, particularly when integrals involve periodic functions which often simplify through these functions.

• In the given integral, the cosine function is key. When dealing with an integral involving \(\cos(u)\), knowing its antiderivative \(\sin(u)\) is crucial.
• These functions are periodic and often evaluate to neat values at certain limits, such as at integer multiples of \(\pi\).
• In this case, the limits \(0\) to \(2\pi^5\) simplify the solution because both \(\sin(2\pi^5)\) and \(\sin(0)\) equal zero, leading to the final integral result of zero.

Understanding these properties of trigonometric functions can turn complex integrals into manageable computations.