\( \int_{0}^{\frac{5}{6}} \frac{\mathrm{dx}}{36 \mathrm{x}^{2}+25} \) Click the icon to view the table of general integration

Calculate or simplify the expression \( \int_{0}^{\frac{5}{6}} \frac{1}{36x^{2}+25}dx \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int_{0}^{\frac{5}{6}} \frac{1}{36x^{2}+25} dx\) - step1: Evaluate the integral: \(\int \frac{1}{36x^{2}+25} dx\) - step2: Transform the expression: \(\int \frac{1}{25\tan^{2}\left(t\right)+25}\times \frac{5}{6}\sec^{2}\left(t\right) dt\) - step3: Simplify the expression: \(\int \frac{\sec^{2}\left(t\right)}{30\tan^{2}\left(t\right)+30} dt\) - step4: Simplify the expression: \(\int \frac{1}{30} dt\) - step5: Evaluate the integral: \(\frac{1}{30}t\) - step6: Substitute back: \(\frac{1}{30}\arctan\left(\frac{6}{5}x\right)\) - step7: Return the limits: \(\left(\frac{1}{30}\arctan\left(\frac{6}{5}x\right)\right)\bigg |_{0}^{\frac{5}{6}}\) - step8: Calculate the value: \(\frac{\pi }{120}\) The integral of \( \frac{1}{36x^{2}+25} \) from 0 to \( \frac{5}{6} \) is \( \frac{\pi}{120} \).