\( \int \frac { \sqrt { 4 x ^ { 2 } - 9 } } { x ^ { 3 } } d x , x = \frac { 3 } { 2 } \)

Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{\sqrt{4x^{2}-9}}{x^{3}} dx\) - step1: Transform the expression: \(\int \frac{8\sqrt{\sec^{2}\left(t\right)-1}}{9\sec^{3}\left(t\right)}\times \frac{3}{2}\sec\left(t\right)\tan\left(t\right) dt\) - step2: Simplify the expression: \(\int \frac{4\sqrt{\sec^{2}\left(t\right)-1}\times \tan\left(t\right)}{3\sec^{2}\left(t\right)} dt\) - step3: Simplify the expression: \(\int \frac{2}{3}\left|\tan\left(t\right)\right|\times \sin\left(2t\right) dt\) - step4: Remove the absolute value bars: \(\int \frac{2}{3}\tan\left(t\right)\sin\left(2t\right) dt\) - step5: Simplify the expression: \(\int \frac{4}{3}\sin^{2}\left(t\right) dt\) - step6: Use properties of integrals: \(\frac{4}{3}\times \int \sin^{2}\left(t\right) dt\) - step7: Evaluate the integral: \(\frac{4}{3}\left(\frac{t}{2}-\frac{\sin\left(2t\right)}{4}\right)\) - step8: Simplify the expression: \(\frac{4}{3}\left(\frac{t}{2}-\frac{1}{4}\sin\left(2t\right)\right)\) - step9: Use the distributive property: \(\frac{4}{3}\times \frac{t}{2}+\frac{4}{3}\left(-\frac{1}{4}\sin\left(2t\right)\right)\) - step10: Multiply the terms: \(\frac{2t}{3}+\frac{4}{3}\left(-\frac{1}{4}\sin\left(2t\right)\right)\) - step11: Multiply the terms: \(\frac{2t}{3}-\frac{1}{3}\sin\left(2t\right)\) - step12: Substitute back: \(\frac{2\operatorname{arcsec}\left(\frac{2}{3}x\right)}{3}-\frac{1}{3}\sin\left(2\operatorname{arcsec}\left(\frac{2}{3}x\right)\right)\) - step13: Simplify the expression: \(\frac{2}{3}\operatorname{arcsec}\left(\frac{2}{3}x\right)-\frac{1}{3}\sin\left(2\operatorname{arcsec}\left(\frac{2}{3}x\right)\right)\) - step14: Add the constant of integral C: \(\frac{2}{3}\operatorname{arcsec}\left(\frac{2}{3}x\right)-\frac{1}{3}\sin\left(2\operatorname{arcsec}\left(\frac{2}{3}x\right)\right) + C, C \in \mathbb{R}\) Substitute \( x=\frac{3}{2} \) into the expression \( \frac{2}{3}\operatorname{arcsec}(\frac{2}{3}x)-\frac{1}{3}\sin(2\operatorname{arcsec}(\frac{2}{3}x)) + C \). Evaluate the expression by following steps: - step0: Evaluate: \(\frac{2}{3}\operatorname{arcsec}\left(\frac{2}{3}x\right)-\frac{1}{3}\sin\left(2\operatorname{arcsec}\left(\frac{2}{3}x\right)\right)+C\) - step1: Substitute: \(\frac{2}{3}\operatorname{arcsec}\left(\frac{2}{3}\times \frac{3}{2}\right)-\frac{1}{3}\sin\left(2\operatorname{arcsec}\left(\frac{2}{3}\times \frac{3}{2}\right)\right)+C\) - step2: Multiply the numbers: \(\frac{2}{3}\operatorname{arcsec}\left(1\right)-\frac{1}{3}\sin\left(2\operatorname{arcsec}\left(\frac{2}{3}\times \frac{3}{2}\right)\right)+C\) - step3: Multiply the numbers: \(\frac{2}{3}\operatorname{arcsec}\left(1\right)-\frac{1}{3}\sin\left(2\operatorname{arcsec}\left(1\right)\right)+C\) - step4: Calculate: \(\frac{2}{3}\operatorname{arcsec}\left(1\right)-\frac{1}{3}\sin\left(2\times 0\right)+C\) - step5: Multiply: \(\frac{2}{3}\operatorname{arcsec}\left(1\right)-\frac{1}{3}\sin\left(0\right)+C\) - step6: Calculate: \(\frac{2}{3}\times 0-\frac{1}{3}\sin\left(0\right)+C\) - step7: Calculate: \(\frac{2}{3}\times 0-\frac{1}{3}\times 0+C\) - step8: Multiply: \(0-\frac{1}{3}\times 0+C\) - step9: Multiply: \(0+0+C\) - step10: Remove 0: \(C\) To solve the integral \[ \int \frac { \sqrt { 4 x ^ { 2 } - 9 } } { x ^ { 3 } } d x, \] we first find the indefinite integral. The result is: \[ \frac{2}{3}\operatorname{arcsec}\left(\frac{2}{3}x\right) - \frac{1}{3}\sin\left(2\operatorname{arcsec}\left(\frac{2}{3}x\right)\right) + C, \] where \( C \) is the constant of integration. Next, we substitute \( x = \frac{3}{2} \) into the expression: \[ \frac{2}{3}\operatorname{arcsec}\left(\frac{2}{3} \cdot \frac{3}{2}\right) - \frac{1}{3}\sin\left(2\operatorname{arcsec}\left(\frac{2}{3} \cdot \frac{3}{2}\right)\right) + C. \] Calculating \( \frac{2}{3} \cdot \frac{3}{2} \) gives us \( 1 \). The arcsecant of \( 1 \) is \( 0 \), so we have: \[ \frac{2}{3} \cdot 0 - \frac{1}{3} \sin(0) + C = C. \] Thus, the final result of the integral evaluated at \( x = \frac{3}{2} \) is simply \( C \).