/*! 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 48 Evaluate the definite integral. ... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 48

Evaluate the definite integral. $$ \int_{2}^{4} \sqrt{3+x^{2}} d x $$

Short Answer

Expert verified
The result of the integral is \( 2\sqrt{3} [\frac{1}{2} (\sec(4)\tan(4) + \ln | \sec(4) + \tan(4) |) - \frac{1}{2} (\sec(2)\tan(2) + \ln | \sec(2) + \tan(2) |)] \)

Step by step solution

01

Trigonometric Substitution

Assign \( x = \sqrt{3} \tan(\theta) \) which implies \( dx = \sqrt{3} \sec^2(\theta) d \theta \). This substitution is suggested because the square root in the integrand will simplify.
02

Substitute and Simplify

Substitute \( x = \sqrt{3} \tan(\theta) \) and \( dx = \sqrt{3} \sec^2(\theta) d \theta \) into the integral: \( \int_{2}^{4} \sqrt{3 + x^2} d x = \int_{\arctan(2 / \sqrt{3})}^{\arctan(4 / \sqrt{3})}\sqrt{3 + 3 \tan^2(\theta)} \sqrt{3} \sec^2(\theta) d \theta \). Then, simplify the integrand, which becomes \( 2\sqrt{3} \int_{\arctan(2 / \sqrt{3})}^{\arctan(4 / \sqrt{3})} \sec^3(\theta) d \theta \)
03

Evaluate the Integral

This integral is now a standard one that is evaluated as follows: \( 2\sqrt{3} [\frac{1}{2} (\sec(\theta)\tan(\theta) + \ln | \sec(\theta) + \tan(\theta) |)]_{\arctan(2 / \sqrt{3})}^{\arctan(4 / \sqrt{3})} \)
04

Substitute the Limits

Substitute back the values using the arctan limits to the above result: \( 2\sqrt{3} [\frac{1}{2} (\sec(\theta)\tan(\theta) + \ln | \sec(\theta) + \tan(\theta) |)]_{2}^{4} \)
05

Simplify the final result

By simplifying, the final answer would be \( 2\sqrt{3} [\frac{1}{2} (\sec(4)\tan(4) + \ln | \sec(4) + \tan(4) |) - \frac{1}{2} (\sec(2)\tan(2) + \ln | \sec(2) + \tan(2) |)] \)

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