Chapter 5: Q. 3. (page 451)

In Example 1(a) we showed that $\int \sin^{3} x \cos^{4} x d x = - \frac{1}{5} \cos^{5} x + \frac{1}{7} \cos^{7} x + C$ . 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

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

Step 2. Differentiation.

By differentiating the function, we get

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

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

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

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In Example 1(a) we showed that $\int \sin^{3} x \cos^{4} x d x = - \frac{1}{5} \cos^{5} x + \frac{1}{7} \cos^{7} x + C$ . 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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In Example 1(a) we showed that ∫sin3xcos4xdx=−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

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

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