Problem 73 - University Calculus: Early Transcendentals

Question

Verify the formulas in Exercises by differentiation. $$\int \sec ^{2}(5 x-1) d x=\frac{1}{5} \tan (5 x-1)+C$$

Step-by-Step Solution

Answer

Yes, the formula is verified by differentiation.

Step 1: Differentiate the Right Side

Let's start by differentiating the right side of the equation to verify it's the antiderivative of the left side. The right side is \( \frac{1}{5} \tan (5x - 1) + C \). Use the chain rule for differentiation. Differentiate \( \tan (u) \) as \( \sec^2(u) \) and \( u = 5x - 1 \), differentiate \( u \) as 5. So,\[\frac{d}{dx}\left(\frac{1}{5} \tan (5x-1) + C\right) = \frac{1}{5} \cdot 5 \cdot \sec^2(5x-1) = \sec^2(5x-1)\]

Step 2: Verify the Derivative Matches the Left Side

Verify that the result from differentiating the right side matches the left side of the original equation \( \sec^2(5x-1) \). Since we calculated:\[\frac{d}{dx}\left(\frac{1}{5} \tan (5x-1) + C\right) = \sec^2(5x-1)\]This matches with the left side \( \sec^2(5x-1) \), confirming the original equation is correctly stated.

Step 3: Conclusion

Since the differentiation of the right side results in the left side, the formula is verified.

Key Concepts

Antiderivatives

Antiderivatives, often referred to as indefinite integrals, are fundamental components of integral calculus. An antiderivative of a function is another function whose derivative is the original function. For example, when you integrate \( \sec^2(5x-1) \), you are looking for a function whose derivative would return \( \sec^2(5x-1) \). This process is essentially reversing differentiation.When you solve an indefinite integral, you should always add a constant \( C \), referred to as the integration constant. This is because differentiation of a constant yields zero, meaning multiple functions can have the same derivative. For the integral \( \int \sec^2(5x-1) \, dx \), the antiderivative we find is \( \frac{1}{5} \tan(5x-1) + C \). This result tells us that differentiating \( \frac{1}{5} \tan(5x-1) + C \) will give us back the original function \( \sec^2(5x-1) \).

Chain Rule

The Chain Rule is a crucial derivative rule used to differentiate composite functions. It allows you to find the derivative of functions where one function is inside another, such as \( \tan(5x-1) \).When applying the chain rule, you differentiate the outer function while keeping the inner function intact, and then multiply by the derivative of the inner function. In our exercise, the outer function is \( \tan(u) \), with the inner function being \( u = 5x - 1 \).Here's how it works for our example:

• Differentiating \( \tan(u) \) gives \( \sec^2(u) \).
• The derivative of \( u = 5x - 1 \) is 5.
• Putting it together, the chain rule gives us: \( \frac{d}{dx}[ \frac{1}{5} \tan(5x-1)] = \frac{1}{5} \times 5 \times \sec^2(5x-1) = \sec^2(5x-1) \).

This correct application of the chain rule verifies our integration result.

Differentiation Verification

Differentiation verification is a vital process in calculus used to confirm the correctness of antiderivatives. It involves taking the derivative of the result of an integration problem to ensure it returns to the original function you integrated.In this exercise, you checked the antiderivative \( \frac{1}{5} \tan(5x-1) + C \) by differentiating it and ensuring your result is \( \sec^2(5x-1) \), the function you originally integrated. By differentiating step-by-step:

• The derivative of \( \tan(5x-1) \) was calculated using the chain rule.
• The result, \( \sec^2(5x-1) \), matched the left side of the equation.

Thus, verification shows that the integral is correctly stated.

Integral Calculus

Integral calculus is the branch of mathematics concerned with the processes of integration and finding antiderivatives. It has numerous applications in physics, engineering, and statistics, often involving calculations of area, volume, or total accumulated quantity over an interval.This particular exercise demonstrates a basic integration problem where you verify the antiderivative by differentiation. The integral \( \int \sec^2(5x-1) \, dx \) produces \( \frac{1}{5} \tan(5x-1) + C \).Integral calculus revolves largely around the relationship between differentiation and integration, which are inverse processes. Understanding this reciprocal relationship allows for the solving of complex problems and is foundational for areas like differential equations, where calculating accumulation is crucial.