/*! 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 32 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 32

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\pi / 4} 2 \cos x d x$$

Short Answer

Expert verified
Question: Evaluate the definite integral: $$\int_{0}^{\pi / 4} 2 \cos x dx$$ Answer: The evaluated definite integral is: $$\sqrt{2}$$

Step by step solution

01

Find the Antiderivative of the Function

To find the antiderivative of the function 2cos(x), we need to find a function F(x) such that: $$\frac{dF(x)}{dx} = 2 \cos x$$ The antiderivative of cos(x) is sin(x). Therefore, the antiderivative of 2cos(x) is 2sin(x). So, we have F(x) = 2sin(x).
02

Apply the Fundamental Theorem of Calculus

Now, we will use the Fundamental Theorem of Calculus to evaluate the definite integral. The theorem states: $$\int_{a}^{b}f(x)dx = F(b) - F(a)$$ In this case, our lower limit a = 0 and upper limit b = 蟺/4, and F(x) = 2sin(x). Therefore, we'll get: $$\int_{0}^{\pi / 4} 2 \cos x d x = F(\pi / 4) - F(0)$$ Now, substitute the values of a and b into F(x) and compute the difference: $$= 2 \sin(\pi / 4) - 2 \sin(0)$$ Since sin(蟺/4) = 鈭�2 / 2 and sin(0) = 0, the result is: $$= 2 \times (\sqrt{2} / 2) - 2 \times 0 = \sqrt{2}$$ Thus, the evaluated definite integral is: $$\int_{0}^{\pi / 4} 2 \cos x dx = \sqrt{2}$$

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Most popular questions from this chapter

Consider the function g. which is given in terms of a definite integral with a variable upper limit. a. Graph the integrand. b. Calculate \(g^{\prime}(x)\) c. Graph g, showing all your work and reasoning. $$g(x)=\int_{0}^{x}\left(t^{2}+1\right) d t$$

Substitutions Suppose that \(f\) is an even integrable function with \(\int_{0}^{8} f(x) d x=9\) a. Evaluate \(\int_{-1}^{1} x f\left(x^{2}\right) d x\) b. Evaluate \(\int_{-2}^{2} x^{2} f\left(x^{3}\right) d x\)

Simplify the given expressions. $$\frac{d}{d x} \int_{0}^{\cos x}\left(t^{4}+6\right) d t$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{8} \sqrt[3]{y} d y$$

Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\) and interpret the results using the graphs of part \((b)\) $$f(x)=-12 x(x-1)(x-2) ; a=0, b=1, c=2$$
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