/*! 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 41 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 41

Evaluate the following integrals or state that they diverge. $$\int_{0}^{1} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int_{0}^{1} \frac{e^{\sqrt{x}}}{\sqrt{x}} dx$$ or state if it diverges. Answer: The integral converges and its value is $$2(e - 1)$$

Step by step solution

01

Substitution

Let \(u = \sqrt{x}\). Then, \(x = u^2\) and the \(dx = 2udu\). Now, we will change the limits of integration accordingly: When \(x = 0\), then \(u = \sqrt{0} = 0\). When \(x = 1\), then \(u = \sqrt{1} = 1\). Now, we can rewrite the integral in terms of \(u\): $$\int_{0}^{1} \frac{e^{\sqrt{x}}}{\sqrt{x}} dx = \int_{0}^{1} \frac{e^u}{u} (2udu) = 2 \int_{0}^{1} e^u du$$
02

Evaluate the integral

Now, we can easily evaluate the integral of \(2 \int_{0}^{1} e^u du\) by finding the antiderivative of the integrand, which is \(2e^u\): $$2 \int_{0}^{1} e^u du = 2 \left[ e^u \right]_{0}^{1} = 2(e^{1} - e^{0}) = 2(e - 1)$$ So, the given integral converges and its value is: $$\int_{0}^{1} \frac{e^{\sqrt{x}}}{\sqrt{x}} dx = 2(e - 1)$$

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