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Chapter 5: Problem 23

Evaluate the integral and check your answer by differentiating. $$\int\left[3 \sin x-2 \sec ^{2} x\right] d x$$

Short Answer

Expert verified
The integral is \(-3 \cos x - 2 \tan x + C\).

Step by step solution

01

Identify the Integral Components

The given integral is \( \int (3 \sin x - 2 \sec^2 x) \, dx \). Identify the two components of the integral: \( 3 \sin x \) and \( -2 \sec^2 x \). Each component will be integrated separately.
02

Integrate the First Component

Integrate \( 3 \sin x \) with respect to \( x \). The antiderivative of \( \sin x \) is \( -\cos x \). Thus, we have:\[\int 3 \sin x \, dx = 3(-\cos x) = -3 \cos x.\]
03

Integrate the Second Component

Integrate \( -2 \sec^2 x \) with respect to \( x \). The antiderivative of \( \sec^2 x \) is \( \tan x \). So, we have:\[\int -2 \sec^2 x \, dx = -2 \cdot \tan x = -2 \tan x.\]
04

Combine the Antiderivatives

Combine the results of Steps 2 and 3 to find the antiderivative of the entire expression:\[\int (3 \sin x - 2 \sec^2 x) \, dx = -3 \cos x - 2 \tan x + C,\]where \( C \) is the constant of integration.
05

Differentiate to Verify

Differentiate the combined result to verify accuracy:\[\begin{align*}\frac{d}{dx}(-3 \cos x - 2 \tan x + C) &= 3 \sin x - 2 \sec^2 x. \end{align*}\]The differentiation matches the original integrand, thus confirming the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration Techniques
When solving integrals like \( \int (3 \sin x - 2 \sec^2 x) \, dx \), it's helpful to identify different integration techniques. In this example, the integral can be broken down into simpler parts: \(3 \sin x\) and \(-2 \sec^2 x\). This technique is known as "separation" or "linear property of integration," and it allows us to solve each part individually.

Other integration techniques can include:
  • Substitution, where you substitute a part of the integral with a new variable to simplify the expression.
  • Integration by parts, useful when an integral is the product of two functions.
  • Partial fraction decomposition, which is handy when dealing with rational functions.
In this case, we've chosen components that have straightforward antiderivatives, making the process quick and efficient. Remember, mastering these techniques gives you flexibility in solving a variety of integrals.
Antiderivative
The goal in integration is to find the antiderivative, which is a function whose derivative equals the integrand. In our example,\[\int (3 \sin x - 2 \sec^2 x) \, dx = -3 \cos x - 2 \tan x + C,\]the antiderivative is the expression \(-3 \cos x - 2 \tan x + C\). Here, the process involves recognizing common antiderivatives:
  • The antiderivative of \( \sin x \) is \(-\cos x\), which is why \( 3 \sin x \) becomes \(-3 \cos x\).
  • The antiderivative of \( \sec^2 x \) is \( \tan x \), leading \(-2 \sec^2 x\) to become \(-2 \tan x\).
Lastly, remember to add \( C \), the constant of integration, which represents any constant offset in the family of antiderivatives. This constant is crucial as it accounts for all possible functions that would differentiate to the integrand. Practicing these will make finding antiderivatives intuitive.
Verification by Differentiation
Once you've computed an antiderivative, it's important to verify your result through differentiation. This step ensures your solution is correct.

From our antiderivative, \(-3 \cos x - 2 \tan x + C\), differentiate to check:
  • The derivative of \(-3 \cos x\) should match \(3 \sin x\), since \(\frac{d}{dx}(-\cos x) = \sin x\).
  • The derivative of \(-2 \tan x\) should return \(-2 \sec^2 x\), given that \(\frac{d}{dx}(\tan x) = \sec^2 x\).
  • The derivative of the constant \(C\) is zero, as constants do not affect the derivative.
Combine these, and you retrieve the original integrand: \(3 \sin x - 2 \sec^2 x\). This successful differentiation process confirms the correctness of our integration. Incorporating this check solidifies understanding and helps prevent errors in your calculus work.

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Most popular questions from this chapter

A function \(f(x)\) is defined piecewise on an interval. In these exercises: (a) Use Theorem 5.5 .5 to find the integral of \(f(x)\) over the interval. (b) Find an anti-derivative of \(f(x)\) on the interval. (c) Use parts (a) and (b) to verify Part 1 of the Fundamental Theorem of Calculus. $$f(x)=\left\\{\begin{array}{ll} \sqrt{x}, & 0 \leq x<1 \\ 1 / x^{2}, & 1 \leq x \leq 4 \end{array}\right.$$

Sketch the curve and find the total area between the curve and the given interval on the \(x\) -axis. $$y=\frac{x^{2}-1}{x^{2}} ;\left[\frac{1}{2}, 2\right]$$

A traffic engineer monitors the rate at which cars enter the main highway during the afternoon rush hour. From her data she estimates that between 4: 30 PM. and 5: 30 P.M. the rate \(R(t)\) at which cars enter the highway is given by the formula \(R(t)=100\left(1-0.0001 t^{2}\right)\) cars per minute, where \(t\) is the time (in minutes) since 4: 30 PM. (a) When does the peak traffic flow into the highway occur? (b) Estimate the number of cars that enter the highway during the rush hour.

Find a positive value of \(k\) such that the area under the graph of \(y=e^{2 x}\) over the interval \([0, k]\) is 3 square units.

$$\begin{aligned} \sum_{k=1}^{4}\left[(k+1)^{3}-k^{3}\right]=&\left[5^{3}-4^{3}\right]+\left[4^{3}-3^{3}\right] \\\ &+\left[3^{3}-2^{3}\right]+\left[2^{3}-1^{3}\right] \\ =& 5^{3}-1^{3}=124 \end{aligned}$$ For convenience, the terms are listed in reverse order. Note how cancellation allows the entire sum to collapse like a telescope. A sum is said to telescope when part of each term cancels part of an adjacent term, leaving only portions of the first and last terms uncanceled. Evaluate the telescoping sums in these exercises. $$\sum_{k=1}^{100}\left(2^{k+1}-2^{k}\right)$$
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