Problem 43 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(3 t^{2}+\sec ^{2} 2 t\right) d t$$

Step-by-Step Solution

Answer

Question: Find the antiderivative of the function $$3t^2 + \sec^2(2t)$$ and check your work by taking the derivative of the antiderivative. Answer: The antiderivative of the function $$3t^2 + \sec^2(2t)$$ is $$t^3 + \frac{1}{2} \tan(2t) + C$$.

Step 1: Find the antiderivative of 3t^2

To find the antiderivative of $$3t^2$$, we need to apply the power rule for integration: $$\int x^n dx = \frac{x^{n+1}}{n+1}$$ Applying this to $$3t^2$$, our integral would be as follows: $$\int 3t^2 dt = 3 \cdot \frac{t^{2+1}}{2+1} = t^3$$

Step 2: Find the antiderivative of sec^2(2t)

Recall that the derivative of the tangent function is the secant squared function: $$\frac{d}{dt}(\tan x) = \sec^2 x$$ This implies that the antiderivative of $$\sec^2(2t)$$ can be expressed using the tangent function. To do this, we will use substitution. Let $$u = 2t$$, then $$du = 2 dt$$ or $$\frac{1}{2} du = dt$$ Now, we can rewrite our integral as follows: $$\int \sec^2(2t) dt = \int \sec^2(u) \cdot \frac{1}{2} du$$ Using the fact that the antiderivative of $$\sec^2(u)$$ is $$\tan(u)$$, we will now calculate our integral: $$\int \sec^2(u) \cdot \frac{1}{2} du = \frac{1}{2} \tan(u)$$ Now, substitute back $$u = 2t$$: $$\frac{1}{2} \tan(2t)$$

Step 3: Combine the antiderivatives

Now that we found the antiderivatives of both terms, we will combine them and add the constant of integration $$C$$: $$\int\left(3 t^{2}+\sec ^{2} 2 t\right) d t = t^3 + \frac{1}{2} \tan(2t) + C$$

Step 4: Check the result by differentiation

We will now take the derivative of our result to make sure it matches the original function. $$\frac{d}{dt}(t^3 + \frac{1}{2} \tan(2t) + C) = 3t^2 + \frac{1}{2} \cdot 2 \sec^2(2t) = 3t^2 + \sec^2(2t)$$ This result matches the original function, so we have found the correct antiderivative.

Key Concepts

Power Rule for Integration

Understanding the power rule for integration is a key step in solving integration problems involving polynomial expressions. The power rule states that the integral of a function of the form \(x^n\) with respect to \(x\) is given by \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\] where \(n\) is not equal to -1 and \(C\) is the constant of integration.

In the given exercise, we applied this rule to find the integral of \(3t^2\). Since we're given \(3t^2\), we treat this as \(3 imes t^2\). Applying the power rule here involves increasing the exponent by one, turning \(t^2\) into \(t^3\), and then dividing by the new exponent, resulting in \(t^3\). Don't forget to include the constant of integration \(C\) at the end of any indefinite integral. A quick note: although the original step by step solution did not explicitly show adding \(C\) after each integration, it mentioned it in the final result.

Substitution Method

The substitution method, sometimes referred to as 'u-substitution,' is a crucial technique in integration when dealing with more complicated expressions. It simplifies the integral by temporarily replacing part of the function with a single variable \(u\).

For instance, in this exercise, we encountered \( \sec^2(2t) \), which can be tricky to integrate directly. By letting \( u = 2t \), and then finding \( du = 2\,dt \) or equivalently \( dt = \frac{1}{2} du \), we transformed our integral into a simpler form \( \int \sec^2(u) \cdot \frac{1}{2} \, du \).

This substitution streamlines the original problem, turning it into a familiar integral \(\int \sec^2(u)\, du\), which we know to be \(\tan(u)\). Converting back to the original variable by substituting \(u = 2t\) gives the antiderivative \(\frac{1}{2} \tan(2t)\). The substitution method is invaluable as it often transforms seemingly complex integrals into more recognizable forms.

Antiderivative

An antiderivative is essentially the reverse process of differentiation, often described as finding the original function given its derivative. In the context of integration, when we compute an indefinite integral, we are essentially finding an antiderivative.

In the exercise provided, the task was to find the antiderivative of \(3t^2 + \sec^2(2t)\). We solved this by first identifying simpler antiderivatives for each part: \(3t^2\) using the power rule, which resulted in \(t^3\), and \(\sec^2(2t)\) using trigonometric identities and substitution, which resulted in \(\frac{1}{2}\tan(2t)\).

By combining these results and remembering to include the constant of integration \(C\), we determined the overall antiderivative as \(t^3 + \frac{1}{2} \tan(2t) + C\). Checking this solution by differentiating it revealed that we returned to the original expression, thus confirming the accuracy of our antiderivative. Mastery of finding antiderivatives is foundational for calculus, especially in solving real-world problems where integration is involved.