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Chapter 7: Problem 1

Evaluate the integral. $$\int \cos ^{3} x \sin x d x$$

Short Answer

Expert verified
\(-\frac{\cos^4 x}{4} + C\)

Step by step solution

01

Identify the Integral Form

The integral given is \( \int \cos ^{3} x \sin x \, dx \). Notice that this form suggests a substitution method could be useful, particularly because the derivative of \( \cos x \) is \( -\sin x \). This indicates that substituting for \( \cos x \) might simplify the integral.
02

Make the Substitution

Let \( u = \cos x \). Then, the derivative is \( du = -\sin x \, dx \), or equivalently, \( -du = \sin x \, dx \). Substitute \( u \) and \( du \) into the integral: \( \int u^3 (-du) \). This simplifies to \( -\int u^3 \, du \).
03

Integrate with Respect to \( u \)

Now, integrate \( -\int u^3 \, du \). Using the power rule for integration, we find: \(-\frac{u^4}{4} + C\).
04

Substitute Back in Terms of \( x \)

Remember that we set \( u = \cos x \). Substitute back to get the solution in terms of \( x \): \[-\frac{(\cos x)^4}{4} + C.\] This can also be written as \(-\frac{\cos^4 x}{4} + C\).

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Key Concepts

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

Substitution Method
The substitution method is a powerful technique for simplifying integrals. It involves changing variables to make integration easier. In essence, we substitute a complicated portion of the integral with a single variable to simplify the process. Here's how it works:

  • **Identify a Substitution:** Look for a part of the integral where a simple derivative is present. For the integral \( \int \cos^3 x \sin x \, dx \), the derivative of \( \cos x \) is \( -\sin x \), makiing it a perfect candidate for substitution.
  • **Substitute the Variable:** Let \( u = \cos x \). Consequently, \( du = -\sin x \, dx \). This transforms our integral to \( \int u^3 (-du) \) or \( -\int u^3 \, du \).
Substitution not only simplifies the integral but also makes the integration process straightforward. The key is to choose the right substitution that reduces the complexity of the given integral.
Trigonometric Integration
Trigonometric integration deals specifically with integrals involving trigonometric functions. These functions, like \( \sin x \) and \( \cos x \), are common in calculus and have special techniques to solve their integrals efficiently.

In our exercise, \( \cos^3 x \sin x \, dx \) involves both \( \cos x \) and \( \sin x \). The choice of substitution (\( u = \cos x \)) effectively transforms the trigonometric components into a simpler polynomial form in \( u \):

  • First, recognize the interplay between \( \cos x \) and \( \sin x \). Since the derivative of \( \cos x \) is \( -\sin x \), this guides us towards the appropriate substitution.
  • Next, use the trigonometric identity \( \sin^2 x + \cos^2 x = 1 \) if needed, though in this case, substitution was enough to simplify the integration.
Trigonometric integration often involves recognizing patterns or applying known identities to streamline solving these integrals.
Power Rule for Integration
The power rule for integration is a fundamental rule in calculus used to integrate functions of the form \( x^n \). The rule states:

If \( n eq -1 \), the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \).

In this example, once we used substitution to transform the integral \( -\int u^3 \, du \), the power rule was directly applicable:

  • **Apply the Power Rule:** Integrate \( u^3 \) to get \( \frac{u^{4}}{4} + C \). Remember the negative outside the integral, leading to the add-on of a negative sign, resulting in \( -\frac{u^4}{4} + C \).
  • **Re-substitute:** Don't forget to re-substitute \( u = \cos x \) back into the integral, yielding the final result: \( -\frac{(\cos x)^4}{4} + C \).
Mastering the power rule is indispensable for evaluating a wide range of integrals, making it a crucial tool in any calculus student's kit.

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Most popular questions from this chapter

(a).Make an appropriate \(u\) -substitution of the form \(u=x^{1 / 2}\) or \(u=(x+a)^{1 / n},\) and then evaluate the integral. (b).If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int \frac{d x}{x^{1 / 2}-x^{1 / 3}}$$

Medication can be administered to a patient using a variety of methods. For a given method, let \(c(t)\) denote the concentration of medication in the patient's bloodstream (measured in \(\mathrm{mg} / \mathrm{L}\) ) \(t\) hours after the dose is given. The area under the curve \(c=c(t)\) over the time interval \([0,+\infty)\) indicates the "availability" of the medication for the patient's body. Determine which method provides the greater availability. Method \(1: c_{1}(t)=6\left(e^{-0.4 t}-e^{-1.3 t}\right)\) Method \(2: c_{2}(t)=5\left(e^{-0.4 t}-e^{-3 t}\right)\)

Evaluate the integral. $$\int \cot ^{3} x \csc ^{3} x d x$$

Find the area of the region between the \(x\) -axis and the curve \(y=8 /\left(x^{2}-4\right)\) for \(x \geq 4\).

Suppose that during a period \(t_{0} \leq t \leq t_{1}\) years, a company has a continuous income stream at a rate of \(I(t)\) dollars per year at time \(t\) and that this income is invested at an annual rate of \(r \%,\) compounded continuously. The value (in dollars) of this income stream at the end of the time period \(t_{0} \leq t \leq t_{1},\) called the stream's future value, can be calculated using $$F V=\int_{t_{0}}^{t_{1}} I(t) e^{r\left(t_{1}-t\right)} d t$$ The present value (in dollars) of the income stream is given by $$P V=\int_{t_{0}}^{t_{1}} I(t) e^{-r\left(t-t_{0}\right)} d t$$ The present value is the amount that, if put in the bank at time \(t=t_{0}\) at \(r \%\) compounded continuously, with no additional deposits, would result in a balance of \(F V\) dollars at time \(t=t_{1}\) That is, $$F V=P V \times e^{r\left(t_{1}-t_{0}\right)}$$ In each exercise, (a) find the future value \(F V\) for the given income stream \(I(t)\) and interest rate \(r\) and time period \(t_{0} \leq t \leq t_{1}\) (b) find the present value \(P V\) of the income stream over the time period; and (c) verify that \(F V\) and \(P V\) satisfy the relationship given above. $$I(t)=2000 t+400 e^{-t} ; r=8 \% ; 0 \leq t \leq 25$$
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