(e) Use the substitution x=3sin theta to evaluate 4 int_(0)^((3)/(sqrt2))(dx)/((9-x^(2))^((3)/(2)))

Apply Substitution: Let's start by applying the substitution x = 3sin(theta). We need to find the corresponding dx and the new limits of integration. dx = 3cos(theta)d(theta)Change Limits and Integration: Now, we substitute x = 3sin(theta) into the integral and change the limits of integration. When x = 0, sin(theta) = 0, so theta = 0. When x = 3/sqrt(2), sin(theta) = 1/sqrt(2), so theta = pi/4.Simplify Denominator: The integral becomes: int_(0)^(pi/4) (3cos(theta)d(theta))/((9 - (3sin(theta))^2)^(3/2)) Simplify the denominator using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1.Further Simplification: The integral simplifies to: int_(0)^(pi/4) (3cos(theta)d(theta))/((9 - 9sin^2(theta))^(3/2)) = int_(0)^(pi/4) (3cos(theta)d(theta))/((9(1 - sin^2(theta)))^(3/2)) = int_(0)^(pi/4) (3cos(theta)d(theta))/((9cos^2(theta))^(3/2))Recognize Integral: Simplify the integral further: = int_(0)^(pi/4) (3cos(theta)d(theta))/(9^(3/2)cos^3(theta)) = int_(0)^(pi/4) (3cos(theta)d(theta))/(27cos^3(theta)) = int_(0)^(pi/4) (1/(9cos^2(theta)))d(theta)Evaluate Antiderivative: Recognize that 1/cos^2(theta) is sec^2(theta), so the integral becomes: = (1/9)int_(0)^(pi/4) sec^2(theta)d(theta)The integral of sec^2(theta) is tan(theta), so we have: = (1/9)[tan(theta)]_(0)^(pi/4)Evaluate the antiderivative at the limits of integration: = (1/9)(tan(pi/4) - tan(0)) = (1/9)(1 - 0) = 1/9

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(e) Use the substitution $\newline$ $x=3\sin \theta$ to evaluate $\newline$ $\int_{0}^{\frac{3}{\sqrt{2}}}\frac{dx}{(9-x^{2})^{\frac{3}{2}}}$

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Find indefinite integrals using the substitution

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Q. (e) Use the substitution

\newline

x=3\sin \theta

to evaluate

\newline

\int_{0}^{\frac{3}{\sqrt{2}}}\frac{dx}{(9-x^{2})^{\frac{3}{2}}}

Apply Substitution: Let's start by applying the substitution $x = 3\sin(\theta)$ . We need to find the corresponding $dx$ and the new limits of integration. $\newline$ $dx = 3\cos(\theta)d(\theta)$
Change Limits and Integration: Now, we substitute $x = 3\sin(\theta)$ into the integral and change the limits of integration. When $x = 0$ , $\sin(\theta) = 0$ , so $\theta = 0$ . When $x = \frac{3}{\sqrt{2}}$ , $\sin(\theta) = \frac{1}{\sqrt{2}}$ , so $\theta = \frac{\pi}{4}$ .
Simplify Denominator: The integral becomes: $\int_{0}^{\pi/4} \frac{3\cos(\theta)d(\theta)}{(9 - (3\sin(\theta))^2)^{3/2}}$ Simplify the denominator using the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ .
Further Simplification: The integral simplifies to: $\int_{0}^{\pi/4} \frac{3\cos(\theta)d(\theta)}{(9 - 9\sin^2(\theta))^{3/2}}$ = $\int_{0}^{\pi/4} \frac{3\cos(\theta)d(\theta)}{(9(1 - \sin^2(\theta)))^{3/2}}$ = $\int_{0}^{\pi/4} \frac{3\cos(\theta)d(\theta)}{(9\cos^2(\theta))^{3/2}}$
Recognize Integral: Simplify the integral further: $\newline$ $= \int_{0}^{\frac{\pi}{4}} \frac{3\cos(\theta)d(\theta)}{9^{\frac{3}{2}}\cos^3(\theta)}$ $\newline$ $= \int_{0}^{\frac{\pi}{4}} \frac{3\cos(\theta)d(\theta)}{27\cos^3(\theta)}$ $\newline$ $= \int_{0}^{\frac{\pi}{4}} \left(\frac{1}{9\cos^2(\theta)}\right)d(\theta)$
Evaluate Antiderivative: Recognize that $\frac{1}{\cos^2(\theta)}$ is $\sec^2(\theta)$ , so the integral becomes: $\newline$ = $(\frac{1}{9})\int_{0}^{\frac{\pi}{4}} \sec^2(\theta)d(\theta)$
Evaluate Antiderivative: Recognize that $\frac{1}{\cos^2(\theta)}$ is $\sec^2(\theta)$ , so the integral becomes: $\newline$ = $(\frac{1}{9})\int_{0}^{\frac{\pi}{4}} \sec^2(\theta)d(\theta)$ The integral of $\sec^2(\theta)$ is $\tan(\theta)$ , so we have: $\newline$ = $(\frac{1}{9})[\tan(\theta)]_{0}^{\frac{\pi}{4}}$
Evaluate Antiderivative: Recognize that $\frac{1}{\cos^2(\theta)}$ is $\sec^2(\theta)$ , so the integral becomes: $\newline$ = $(\frac{1}{9})\int_{0}^{\frac{\pi}{4}} \sec^2(\theta)d(\theta)$ The integral of $\sec^2(\theta)$ is $\tan(\theta)$ , so we have: $\newline$ = $(\frac{1}{9})[\tan(\theta)]_{0}^{\frac{\pi}{4}}$ Evaluate the antiderivative at the limits of integration: $\newline$ = $(\frac{1}{9})(\tan(\frac{\pi}{4}) - \tan(0))$ $\newline$ = $(\frac{1}{9})(1 - 0)$ $\newline$ = $\frac{1}{9}$