Q. 45 Combining derivatives and integr... [FREE SOLUTION]

Chapter 4: Q. 45 (page 405)

Combining derivatives and integrals: Simplify each of the following as much as possible:
$\frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t$

Short Answer

Expert verified

The solution is $\frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t = \frac{3}{4 x} (\sin (\ln x) - \frac{1}{3} \sin 3 (\ln x)) .$

Step by step solution

Step 1. Given information

Integral: $\frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t$

Step 2. Calculation

The given equation is-

$\frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t$

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

$\frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t = \frac{d}{d x} {鈭�}_{0}^{\ln x} (\frac{3}{4} \sin t - \frac{1}{4} \sin 3 t) d t \frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t = \frac{d}{d x} (\frac{3}{4} {鈭�}_{0}^{\ln x} (\sin t) d t - \frac{1}{4} {鈭�}_{0}^{\ln x} (\sin 3 t) d t) U \sin g 鈭� \sin x d x = - \cos x + C \frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t = \frac{d}{d x} (\frac{3}{4} {[- \cos t]}_{0}^{\ln x} - \frac{1}{4} {[- \frac{1}{3} \cos 3 t]}_{0}^{\ln x}) \frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t = \frac{d}{d x} (- \frac{3}{4} [\cos (\ln x) - \cos 0] + \frac{1}{12} [\cos 3 (\ln x) - \cos 0]) \frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t = (- \frac{3}{4} \frac{d}{d x} [\cos (\ln x) - 1] + \frac{1}{12} \frac{d}{d x} [\cos 3 (\ln x) - 1]) U \sin g \frac{d}{d x} \cos x = - \sin x \frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t = (- \frac{3}{4} [- \sin (\ln x) \frac{d}{d x} \ln x - 0] + \frac{1}{12} [- \sin 3 (\ln x) \frac{d}{d x} 3 \ln x]) \frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t = (\frac{3}{4} [\frac{\sin (\ln x)}{x}] - \frac{3}{12} [\frac{\sin (\ln x)}{x}]) \frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t = \frac{3}{4 x} (\sin (\ln x) - \frac{1}{3} \sin 3 (\ln x))$

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Combining derivatives and integrals: Simplify each of the following as much as possible:
$\frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t$

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�0lnxsin3tdt

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

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Combining derivatives and integrals: Simplify each of the following as much as possible:
$\frac{d}{d x} {鈭�}_{0}^{\ln x} \sin^{3} t d t$