/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Find the integral. $$ \int \... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 17

Find the integral. $$ \int \frac{1}{\sqrt{16-x^{2}}} d x $$

Short Answer

Expert verified
\(\arcsin\left(\frac{x}{4}\right) + C\)

Step by step solution

01

Trigonometric substitution

Start with the function \(\int \frac{1}{\sqrt{16-x^{2}}} dx\). In this case, \(a = 4\) because the square root term is in terms of \(16 = 4^2\). So, we make a trigonometric substitution of \(x = 4\sin(\theta)\). Then, differentiate \(x = 4\sin(\theta)\) with respect to \(\theta\) to get \(dx = 4\cos(\theta) d\theta\). Now, replace \(x\) with \(4\sin(\theta)\) in the function and \(dx\) with \(4\cos(\theta) d\theta\). The function will become \(\int \frac{1}{\sqrt{16-(4\sin(\theta))^2}} \cdot 4\cos(\theta) d\theta\).
02

Simplify and integrate

Simplify the function under the square root to get \(\int \frac{1}{\sqrt{16(1-\sin^2(\theta))}} \cdot 4\cos(\theta) d\theta\). Further simplification due to \(1-\sin^2(\theta) = \cos^2(\theta)\) gives \(\int \frac{4\cos(\theta)}{\sqrt{16\cos^2(\theta)}} d\theta\), which further simplifies to \(\int d\theta\). The integral of \(d\theta\) is \(\theta\) + C.
03

Substitute \(\theta\) back in terms of \(x\)

Now replace \(\theta\) with \(\arcsin\left(\frac{x}{4}\right)\) because from the original substitution \(x = 4\sin(\theta)\), \(\theta = \arcsin\left(\frac{x}{4}\right)\). Thus, the final answer is \(\arcsin\left(\frac{x}{4}\right) + C\).

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Most popular questions from this chapter

(a) The improper integrals \(\int_{1}^{\infty} \frac{1}{x} d x \quad\) and \(\int_{1}^{\infty} \frac{1}{x^{2}} d x\) diverge and converge, respectively. Describe the essential differences between the integrands that cause one integral to converge and the other to diverge. (b) Sketch a graph of the function \(y=\sin x / x\) over the interval \((1, \infty)\). Use your knowledge of the definite integral to make an inference as to whether or not the integral \(\int_{1}^{\infty} \frac{\sin x}{x} d x\) converges. Give reasons for your answer. (c) Use one iteration of integration by parts on the integral in part (b) to determine its divergence or convergence.

(A) find the indefinite integral in two different ways. (B) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (C) Verify analytically that the results differ only by a constant. $$ \int \sec ^{4} 3 x \tan ^{3} 3 x d x $$

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=e^{a t} $$

Show that \(\lim _{x \rightarrow \infty} \frac{x^{n}}{e^{x}}=0\) for any integer \(n>0\).

Rewrite the improper integral as a proper integral using the given \(u\) -substitution. Then use the Trapezoidal Rule with \(n=5\) to approximate the integral. $$ \int_{0}^{1} \frac{\sin x}{\sqrt{x}} d x, \quad u=\sqrt{x} $$
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