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Chapter 5: Q 33. (page 452)

Solve the following integral.
$\int \cos^{4} x d x$

Short Answer

Expert verified

Answer is $\frac{\sin 2 x}{4} + \frac{\sin 4 x}{32} + \frac{3 x}{8} + C$

Step by step solution

Step 1. Given information

An integral is $\int \cos^{4} x d x$

Step 2. Explanation

$\begin{array}{rcl} \int \cos^{4} x d x & = & \frac{\cos 2 x}{2} + \frac{\cos 4 x}{8} + \frac{3}{8} \\ = & \frac{\sin 2 x}{4} + \frac{\sin 4 x}{32} + \frac{3 x}{8} + C \end{array}$

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Most popular questions from this chapter

Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial $x^{2} - x + 3$ as the quotient above the top line, and the polynomial 3x − 1 at the bottom as the remainder. Then $p (x) =___a n d \frac{p (x)}{q (x)} =____$

Solve the integral: $\int x^{2} \cos x d x$ .

Which of the integrals that follow would be good candidates for trigonometric substitution? If a trigonometric substitution is a good strategy, name the substitution. If another method is a better strategy, explain that method.

$(a) \int \frac{4 + x^{2}}{x} d x$ $(b) \int \frac{x}{4 + x^{2}} d x$

role="math" localid="1648759296940" $(c) \int \frac{x^{2}}{4 + x^{2}} d x$ $(d) \int \frac{16 - x^{4}}{4 + x^{2}} d x$

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{x^{2} - 4} d x$ .

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving $\int \frac{1}{\sqrt{x^{2} - 4}} d x$ .

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving $\int {(x^{2} + 4)}^{- 5 / 2} d x$

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form $\sqrt{x^{2} - a^{2}}$ .

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with $x = a \sin u$ , we must consider the cases $x > a$ and $x < - a$ separately.

(h) True or False: When using trigonometric substitution with $x = a s e c u$ , we must consider the cases $x > a$ and $x < - a$ separately.

Solve given definite integrals.

$\int_{0}^{4} x \sqrt{x^{2} + 4} d x$

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Solve the following integral.∫cos4xdx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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Most popular questions from this chapter

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Solve the following integral.
$\int \cos^{4} x d x$