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Chapter 8: Problem 7

Evaluate the integrals. $$\int \sin ^{5} x d x$$

Short Answer

Expert verified
\( \int \sin^5(x) \, dx = -\cos(x) + \frac{2}{3}\cos^3(x) - \frac{1}{5}\cos^5(x) + C \).

Step by step solution

01

Use a Power-Reduction Identity

To handle the odd power of the sine function, recall that \( sin^2(x) = 1 - cos^2(x) \). We can rewrite \( sin^5(x) \) as \( sin^4(x) \cdot sin(x) \) then use the identity on \( sin^4(x) \). This gives \( \int sin^5(x) \, dx = \int (sin^2(x))^2 \cdot sin(x) \, dx = \int (1 - cos^2(x))^2 \cdot sin(x) \, dx \).
02

Make a Substitution

Using the substitution \( u = cos(x) \), find \( du = -sin(x)\,dx \), or \( -du = sin(x)\,dx \). Substitute into the integral: \( \int (1 - u^2)^2 \cdot (-du) = \int -(1 - u^2)^2 \, du \).
03

Expand the Expression

Expand \( (1 - u^2)^2 \) to compute it easily: \( 1 - 2u^2 + u^4 \). Substitute this into the integral from Step 2 to get \( \int -(1 - 2u^2 + u^4) \, du = \int (-1 + 2u^2 - u^4) \, du \).
04

Integrate Term by Term

Integrate each term individually: \( \int -1 \, du = -u \), \( \int 2u^2 \, du = \frac{2}{3}u^3 \), and \( \int -u^4 \, du = -\frac{1}{5}u^5 \). Combine them: \( -u + \frac{2}{3}u^3 - \frac{1}{5}u^5 + C \).
05

Substitute Back for the Original Variable

Replace \( u \) with \( cos(x) \): \( -cos(x) + \frac{2}{3}cos^3(x) - \frac{1}{5}cos^5(x) + C \). This is the antiderivative of the original integral.

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Key Concepts

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

Power-Reduction Identity
When dealing with integrals involving odd powers of sine or cosine, the power-reduction identity is incredibly helpful. It allows us to transform the function into something easier to integrate. For sine, we use the identity:
  • \( \sin^2(x) = 1 - \cos^2(x) \)

For our integral \( \int \sin^5(x) \, dx \), we first express \( \sin^5(x) \) as \( \sin^4(x) \cdot \sin(x) \). Breaking \( \sin^4(x) \) further, we utilize the identity and rewrite it as \( (\sin^2(x))^2 \) and then substitute, getting \( (1 - \cos^2(x))^2 \cdot \sin(x) \).
This technique transforms a tricky integral into one that's more manageable, setting us up to use substitution more effectively.
Substitution Method
The substitution method is a powerful integration technique, especially when combined with trigonometric integrals. In the context of our problem, we perform a substitution to simplify the integral. Here鈥檚 the step:
  • Let \( u = \cos(x) \), thus \( du = -\sin(x)\,dx \), and \( -du = \sin(x)\,dx \).

We substitute \( u \) into the integral, changing \( \int (1 - \cos^2(x))^2 \cdot \sin(x)\, dx \) into \( \int -(1 - u^2)^2 \, du \).
This change simplifies the expression significantly, turning it into a polynomial that is much easier to integrate term by term. Through substitution, we go from an involved trigonometric integral into a clean polynomial form.
Trigonometric Integrals
Integrals involving trigonometric functions can sometimes seem complex, but using specific strategies helps in simplifying them. For odd powers, like \( \sin^5(x) \), breaking down the expression and substituting identities are key tactics.
Here's what you can initially attempt:
  • For odd powers of sine or cosine, isolate one sine or cosine to use with du in substitution.
  • Apply trigonometric identities (like power-reduction) to rewrite the rest in terms of cosine or sine.
  • Use the substitution method to transform and simplify the integral.

Understanding these guidelines is crucial for tackling not just sine integrals but other trigonometric expressions. With practice, these methods become intuitive, opening the door to successfully evaluate more complex integrals.

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Most popular questions from this chapter

Many chemical reactions are the result of the interaction of two molecules that undergo a change to produce a new product. The rate of the reaction typically depends on the concentrations of the two kinds of molecules. If \(a\) is the amount of substance \(A\) and \(b\) is the amount of substance \(B\) at time \(t=0,\) and if \(x\) is the amount of product at time \(t,\) then the rate of formation of \(x\) may be given by the differential equation $$\frac{d x}{d t}=k(a-x)(b-x)$$ or $$\frac{1}{(a-x)(b-x)} \frac{d x}{d t}=k$$ where \(k\) is a constant for the reaction. Integrate both sides of this equation to obtain a relation between \(x\) and \(t\) (a) if \(a=b,\) and (b) if \(a \neq b .\) Assume in each case that \(x=0\) when \(t=0\)

Solve the initial value problems in Exercises \(51-54\) for \(x\) as a function of \(t\). $$\left(t^{2}-3 t+2\right) \frac{d x}{d t}=1 \quad(t>2), \quad x(3)=0$$

The function $$\operatorname{erf}(x)=\int_{0}^{x} \frac{2 e^{-t^{2}}}{\sqrt{\pi}} d t$$ called the error function, has important applications in probability and statistics. a. Plot the error function for \(0 \leq x \leq 25\) b. Explore the convergence of $$\int_{0}^{\infty} \frac{2 e^{-t^{2}}}{\sqrt{\pi}} d t$$ If it converges, what appears to be its value? You will see how to confirm your estimate in Section \(14.4,\) Exercise \(41 .\)

Evaluate the integrals in Exercises \(39-50\). $$\int \frac{(x-2)^{2} \tan ^{-1}(2 x)-12 x^{3}-3 x}{\left(4 x^{2}+1\right)(x-2)^{2}} d x$$

In each case, check your work by differentiating your answer with respect to \(x\). $$\int \sinh ^{-1} x d x$$.
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