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Chapter 6: Problem 1

Evaluate the integrals. $$\int \cos x \sin ^{4} x d x$$

Short Answer

Expert verified
The integral of \( \cos(x)\sin^{4}(x) \) with respect to \( x \) is \( -\left[\cos(x) - \frac{2\cos^{3}(x)}{3} + \frac{\cos^{5}(x)}{5}\right] + C \)

Step by step solution

01

Use the power reducing identity

First, let's substitute \( \sin^{2}x \) by using the power reducing identity, so the integral becomes: \[ \int \cos(x) (1 - \cos^{2}(x))^{2} dx \].
02

Expand the Term

Expand the bracket \((1 - \cos^{2}(x))^{2}\) to give the equation \[ \int \cos(x) (1 - 2\cos^{2}(x) + \cos^{4}(x)) dx \]. This way, the integral will be easier to solve using the substitution method.
03

Simple Substitution

Carry out the substitution \( u = \cos(x) \), then \( du = -\sin(x) dx \). So \( -du = \sin(x) dx \). Substitute these into the integral: \[ -\int (1 - 2u^{2} + u^{4}) du \].
04

Integrate

Now integrate each term in the integral separately: \[ -\left[u - \frac{2u^{3}}{3} + \frac{u^{5}}{5}\right] + C \]
05

Reverse Substitution

Finally resubstitute \( u = \cos(x) \) to find the antiderivative: \[ -\left[\cos(x) - \frac{2\cos^{3}(x)}{3} + \frac{\cos^{5}(x)}{5}\right] + C \]

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Key Concepts

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

Power Reducing Identity
When facing integrals involving high powers of sine and cosine, using trigonometric identities can simplify the process. The power reducing identity is especially useful here. It helps convert higher powers into expressions involving lower powers which are more manageable. In this problem, we use the identity for sine:
  • \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \)
By substituting, the given integral becomes easier to manage. Though the exact formula used in the solution is the rewritten form of the squared identity for easier expansion, this substitution simplifies calculations significantly. Expand and simplify the expression before the main calculation begins and makes integration over trigonometric functions achievable.
Substitution Method
The substitution method is a powerful tool used in integration, similar to changing variables. It simplifies the integration process by transforming the integral into a simpler form. In this exercise, we choose a variable substitution that relates closely to our function.

We set:
  • \( u = \cos(x) \)
This results in the differentials becoming a smooth transition:
  • \( du = -\sin(x)dx \) leading to \( -du = \sin(x)dx \)
This step condenses the complexity of trigonometric integrals into polynomial functions, which are typically easier to deal with. Always remember to revert to the original variable after integrating by substituting back the initial expressions.
Integral Evaluation
After applying the substitution, the integral simplifies to a polynomial form:
  • \( - \int (1 - 2u^2 + u^4) du \)
Evaluating integrals like these involves integrating each term independently:
  • \( \int 1 \, du = u \ \int 2u^2 \, du = \frac{2}{3}u^3 \ \int u^4 \, du = \frac{u^5}{5} \)
The powerful simplicity of these calculations highlights the beauty of definite and indefinite integrals within polynomial expressions. Ultimately, sum up these results, re-introduce the original variables, and never forget to add the constant of integration \( C \). Without this step, the integral is incomplete.

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Most popular questions from this chapter

Many probability questions involve conditional probabilities. For example, if you know that a light bulb has already burned for 30 hours, what is the probability that it will last at least 5 more hours? This is the "probability that \(x > 35\) given that \(x > 30 "\) and is written as \(P(x > 35 | x > 30)\). In general, for events \(A\) and \(B, P(A | B)=\frac{P(A \text { and } B)}{P(B)},\) which in this case reduces to \(P(x > 35 | x > 30)=\frac{P(x > 35)}{P(x > 30)} .\) For the pdf \(f(x)=\frac{1}{40} e^{-x / 40}\) (in hours), compute \(P(x > 35 | x > 30) .\) Also, compute \(P(x > 40 | x>35)\) and \(P(x > 45 | x > 40)\). (Hint: \(\left.P(x>35)=1-P(x \leq 35)=1-\int_{0}^{35} f(x) d x .\right)\)

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