Chapter 17: Problem 1

Find $\int \frac{x}{\sqrt{4-x^{2}}} \mathrm{~d} x$.

Short Answer

Expert verified

The final solution of the integral is $\int \frac{x}{\sqrt{4-x^{2}}} \mathrm{~d} x = -\sqrt{4 - x^2} + C$, where C is the constant of integration.

Step by step solution

Choose an appropriate trigonometric substitution

To make the denominator simpler, we can use the identity $1 - \sin^2(\theta) = \cos^2(\theta)$. We can choose a trigonometric substitution such that $4 - x^2 = 4\cos^2(\theta)$. Let $x = 2\sin(\theta)$, then $\mathrm{d} x = 2\cos(\theta) \mathrm{~d}\theta$.

Substitute and simplify the integrand

Now substitute $x = 2\sin(\theta)$ and $\mathrm{d} x = 2\cos(\theta) \mathrm{~d}\theta$ in the integral and simplify the integrand: \begin{align*} \int \frac{x}{\sqrt{4-x^{2}}} \mathrm{~d} x &= \int \frac{2\sin(\theta)}{\sqrt{4 - (2\sin(\theta))^2}} (2\cos(\theta) \mathrm{~d}\theta) \\ &= \int \frac{4\sin(\theta)\cos(\theta)}{\sqrt{4(1 - \sin^2(\theta))}} \mathrm{~d}\theta \\ &= \int \frac{4\sin(\theta)\cos(\theta)}{2\sqrt{\cos^2(\theta)}} \mathrm{~d}\theta \\ &= \int 2\sin(\theta) \mathrm{~d}\theta \end{align*}

Evaluate the integral in terms of $\theta$

The integral has now become simpler. Evaluate the integral with respect to $\theta$. $$ \int 2\sin(\theta) \mathrm{~d}\theta = -2\cos(\theta) + C $$ where C is the constant of integration.

Convert back to $x$

Now we substitute $\theta$ back in terms of $x$. We know $x = 2\sin(\theta)$. Then, $\sin(\theta) = \frac{x}{2}$. Since $\sin^2(\theta) + \cos^2(\theta) = 1$, we can find $\cos(\theta)$: $$ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \left(\frac{x}{2}\right)^2 = \frac{4 - x^2}{4} $$ So, $\cos(\theta) = \frac{\sqrt{4 - x^2}}{2}$. Finally, substitute $\cos(\theta)$ back into the expression for the integral: $$ -2\cos(\theta) + C = -2\left(\frac{\sqrt{4 - x^2}}{2}\right) + C = -\sqrt{4 - x^2} + C $$ Thus, the final solution is: $$ \int \frac{x}{\sqrt{4-x^{2}}} \mathrm{~d} x = -\sqrt{4 - x^2} + C $$

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Find \(\int \frac{x}{\sqrt{4-x^{2}}} \mathrm{~d} x\).

Short Answer

Step by step solution

Choose an appropriate trigonometric substitution

Substitute and simplify the integrand

Evaluate the integral in terms of \(\theta\)

Convert back to \(x\)

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