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Chapter 5: Problem 40

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}$$

Short Answer

Expert verified
Question: Evaluate the definite integral: $$\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}$$ Answer: \(\frac{\pi}{6}\)

Step by step solution

01

Identify the integrand and the limits of integration

The given integral is: $$\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}$$ The integrand is \(\frac{1}{\sqrt{1-x^{2}}}\), and the limits of integration are 0 (lower limit) and \(\frac{1}{2}\) (upper limit).
02

Find the antiderivative of the integrand

The integrand is a standard integral whose antiderivative is known. The antiderivative of \(\frac{1}{\sqrt{1-x^{2}}}\) is \(\sin^{-1}(x)+C\), where C is the constant of integration. In our case, we do not need to worry about the constant of integration since we are evaluating a definite integral.
03

Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, to evaluate the definite integral, we need to find the difference between the antiderivative evaluated at the upper limit and the antiderivative evaluated at the lower limit. $$\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}} = \sin^{-1}(x)\bigg|_0^{1/2}$$
04

Evaluate the antiderivative at the limits of integration

To find the value of the definite integral, plug in the limits of integration into the antiderivative and compute the difference: $$\sin^{-1}\left(\frac{1}{2}\right) - \sin^{-1}(0)$$
05

Calculate the value of the integral

Evaluate the difference to obtain the value of the integral: $$\sin^{-1}\left(\frac{1}{2}\right) - \sin^{-1}(0) = \frac{\pi}{6} - 0 = \frac{\pi}{6}$$ Therefore, the value of the definite integral is \(\frac{\pi}{6}\).

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Most popular questions from this chapter

General results Evaluate the following integrals in which the function \(f\) is unspecified. Note \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f\). Assume \(f\) and its derivatives are continuous for all real numbers. \(\int\left(f^{(p)}(x)\right)^{n} f^{(p+1)}(x) d x,\) where \(p\) is a positive integer, \(n \neq-1\)

Suppose \(f\) is continuous on \([0, \infty)\) and \(A(x)\) is the net area of the region bounded by the graph of \(f\) and the \(t\) -axis on \([0, x] .\) Show that the maxima and minima of \(A\) occur at the zeros of \(f\). Verify this fact with the function \(f(x)=x^{2}-10 x\)

Additional integrals Use a change of variables to evaluate the following integrals. $$\int\left(x^{3 / 2}+8\right)^{5} \sqrt{x} d x$$

Substitution: scaling Another change of variables that can be interpreted geometrically is the scaling \(u=c x,\) where \(c\) is a real number. Prove and interpret the fact that $$\int_{a}^{b} f(c x) d x=\frac{1}{c} \int_{\alpha c}^{b c} f(u) d u$$ Draw a picture to illustrate this change of variables in the case that \(f(x)=\sin x, a=0, b=\pi,\) and \(c=\frac{1}{2}\)

Additional integrals Use a change of variables to evaluate the following integrals. $$\int_{\sqrt{2}}^{\sqrt{3}}(x-1)\left(x^{2}-2 x\right)^{11} d x$$
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