Chapter 5: Q. 3. (page 451)

In Example 1(a) we showed that $鈭� \sin^{3} x \cos^{4} x d x = 鈭� \frac{1}{5} \cos^{5} x + \frac{1}{7} \cos^{7} x + C$ . Check that this answer is correct by differentiating and applying trigonometric identities

Short Answer

Expert verified

$\sin^{3} x \cos^{4} x$ .

Step by step solution

Step 1. Given Information.

The given integral is

$鈭� \sin^{3} x \cos^{4} x d x = 鈭� \frac{1}{5} \cos^{5} x + \frac{1}{7} \cos^{7} x + C$ .

Step 2. Differentiation.

By differentiating the function, we get

$\frac{d}{d x} (鈭� \frac{1}{5} \cos^{5} x + \frac{1}{7} \cos^{7} x + C) = \sin^{3} x \cos^{4} x$

By using the formulae, $\frac{d}{d x} (\cos x) = - \sin x, \frac{d}{d x} (c) = 0, \frac{d}{d x} (x^{n}) = n x^{n - 1}$ .

$\frac{d}{d x} (鈭� \frac{1}{5} \cos^{5} x + \frac{1}{7} \cos^{7} x + C) = \sin^{3} x \cos^{4} x = 鈭� \frac{1}{5} \frac{d}{d x} (\cos^{5} x) + \frac{1}{7} \frac{d}{d x} (\cos^{7} x) + \frac{d}{d x} (c) = 鈭� \frac{1}{5} 路 5 \cos^{4} x 路 (- \sin x) + \frac{1}{7} 路 7 \cos^{6} x (- \sin x) + 0 = \cos^{4} x \sin x - \cos^{6} x \sin x = \cos^{4} x \sin x (1 - \cos^{2} x) [1 = \cos^{2} x + \sin^{2} x] = \cos^{4} x \sin x 路 \sin^{2} x = \cos^{4} x 路 \sin^{3} x$

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91影视

In Example 1(a) we showed that $鈭� \sin^{3} x \cos^{4} x d x = 鈭� \frac{1}{5} \cos^{5} x + \frac{1}{7} \cos^{7} x + C$ . Check that this answer is correct by differentiating and applying trigonometric identities

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Differentiation.

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91影视

In Example 1(a) we showed that 鈭�sin3xcos4xdx=鈭�15cos5x+17cos7x+C. Check that this answer is correct by differentiating and applying trigonometric identities

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Differentiation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Statistics

Mechanics Maths

Theoretical and Mathematical Physics

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

In Example 1(a) we showed that $鈭� \sin^{3} x \cos^{4} x d x = 鈭� \frac{1}{5} \cos^{5} x + \frac{1}{7} \cos^{7} x + C$ . Check that this answer is correct by differentiating and applying trigonometric identities