/*! 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 10 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 10

Evaluate the following integrals. $$\int_{0}^{\sqrt{2}} \frac{x^{2}}{\sqrt{4-x^{2}}} d x$$

Short Answer

Expert verified
Based on the step-by-step solution provided, the evaluated integral of \(\int_{0}^{\sqrt{2}} \frac{x^{2}}{\sqrt{4-x^{2}}} dx\) is \(\pi - 4\).

Step by step solution

01

Perform the appropriate trigonometric substitution

In this case, we can use the sine function, which makes the substitution easier. Let \(x = 2\sin(\theta)\). Consequently, \(dx = 2\cos(\theta) d\theta\). Note that when \(x=0\), \(\theta=0\) and when \(x=\sqrt{2}\), \(\theta=\frac{\pi}{4}\), as \(\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\). Now substitute the values of \(x\) and \(dx\) in the original integral: $$\int_{0}^{\sqrt{2}} \frac{x^{2}}{\sqrt{4-x^{2}}} d x = \int_{0}^{\frac{\pi}{4}} \frac{(2\sin\theta)^{2}}{\sqrt{4-(2\sin\theta)^{2}}} (2\cos\theta) d\theta$$
02

Simplify the integral

Simplify the integral by cancelling out terms: $$\int_{0}^{\frac{\pi}{4}} \frac{4\sin^{2}\theta}{\sqrt{4-4\sin^{2}\theta}} (2\cos\theta) d\theta = \int_{0}^{\frac{\pi}{4}} \frac{4\sin^{2}\theta}{\sqrt{4\cos^{2}\theta}} (2\cos\theta) d\theta = \int_{0}^{\frac{\pi}{4}} 4\sin^{2}\theta d\theta$$
03

Use the double angle formula for sine

Recall the double angle formula for cosine: \(\cos(2\theta) = 1 - 2\sin^{2}\theta\). We can rewrite it as \(\sin^{2}\theta = \frac{1 - \cos(2\theta)}{2}\) to simplify the integral further: $$\int_{0}^{\frac{\pi}{4}} 4\sin^{2}\theta d\theta = \int_{0}^{\frac{\pi}{4}} 4\left(\frac{1-\cos(2\theta)}{2}\right) d\theta = \int_{0}^{\frac{\pi}{4}} (4 - 8\cos(2\theta)) d\theta$$
04

Evaluate the integral

Now, integrate each term with respect to \(\theta\): $$\int_{0}^{\frac{\pi}{4}} (4 - 8\cos(2\theta)) d\theta = \left[ 4\theta - 4\sin(2\theta)\right]_{0}^{\frac{\pi}{4}}$$
05

Apply the limits of integration

Apply the limits of integration to evaluate the integral: $$\left[ 4\theta - 4\sin(2\theta)\right]_{0}^{\frac{\pi}{4}} = \left[ 4\left(\frac{\pi}{4}\right) - 4\sin\left(2\left(\frac{\pi}{4}\right)\right)\right] - \left[ 4(0) - 4\sin(2(0))\right] = \pi - 4$$ So, the evaluated integral is: $$\int_{0}^{\sqrt{2}} \frac{x^{2}}{\sqrt{4-x^{2}}} d x = \pi - 4$$

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