Q. 46 Combining derivatives and integr... [FREE SOLUTION]

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Chapter 4: Q. 46 (page 405)

Combining derivatives and integrals: Simplify each of the following as much as possible:
$\frac{d}{d x} {鈭�}_{x}^{3} \sin^{3} t d t$

Short Answer

Expert verified

The solution is $\frac{d}{d x} {鈭�}_{x}^{3} \sin^{3} t d t = (- \frac{3}{4} [\sin x] + \frac{1}{4} [\sin 3 x])$

Step by step solution

Step 1. Given information

Integral: $\frac{d}{d x} {鈭�}_{x}^{3} \sin^{3} t d t$

Step 2. Calculation

The given integration is $\frac{d}{d x} {鈭�}_{x}^{3} \sin^{3} t d t$

$\frac{d}{d x} {鈭�}_{x}^{3} \sin^{3} t d t$

Using $\sin^{3} x = \frac{3}{4} \sin x - \frac{1}{4} \sin 3 x$

$= \frac{d}{d x} {鈭�}_{x}^{3} (\frac{3}{4} \sin t - \frac{1}{4} \sin 3 t) d t = \frac{d}{d x} 鈭� (\frac{3}{4} {鈭�}_{x}^{3} (\sin t) d t - \frac{1}{4} {鈭�}_{x}^{3} (\sin 3 t) d t) U \sin g 鈭� \sin x d x = - \cos x + C = \frac{d}{d x} (\frac{3}{4} {[- \cos t]}_{x}^{3} - \frac{1}{4} {[- \frac{1}{3} \cos 3 t]}_{x}^{3}) = \frac{d}{d x} (- \frac{3}{4} [\cos 3 - \cos x] + \frac{1}{12} [\cos 9 - \cos 3 x]) = (- \frac{3}{4} \frac{d}{d x} [\cos 3 - \cos x] + \frac{1}{12} \frac{d}{d x} [\cos 9 - \cos 3 x] u \sin g \frac{d}{d x} \cos x = - \sin x = (- \frac{3}{4} [\sin x] + \frac{1}{12} [3 \sin 3 x]) = (\frac{3}{4} [s i n x] + \frac{1}{4} [\sin 3 x])$

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91影视

Combining derivatives and integrals: Simplify each of the following as much as possible:
$\frac{d}{d x} {鈭�}_{x}^{3} \sin^{3} t d t$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

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91影视

Combining derivatives and integrals: Simplify each of the following as much as possible:ddx鈭�x3sin3tdt

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Applied Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Combining derivatives and integrals: Simplify each of the following as much as possible:
$\frac{d}{d x} {鈭�}_{x}^{3} \sin^{3} t d t$