\( \left. \begin{array} { l } { 147 \int _ { \sqrt { 3 } } ^ { 3 } \frac { 1 } { 9 + x ^ { 2 } } d x } \\ { u ^ { 2 } = x ^ { 2 } a ^ { 2 } = 9 \quad a = 3 } \\ { u = x \quad d u = d x } \\ { = \int _ { \sqrt { 3 } } ^ { 3 } \frac { 1 } { a ^ { 2 } + u ^ { 2 } } d u = \frac { 1 } { 3 } \operatorname { Tan } ^ { - 1 } ( \frac { x } { 3 } ) + c ] _ { \sqrt { 3 } } ^ { 3 } } \\ { = \frac { 1 } { 3 } \operatorname { Tan } ^ { - 1 } ( \frac { 3 } { 3 } ) - [ \frac { 1 } { 3 } \operatorname { Tan } ^ { - 1 } ( \frac { \sqrt { 3 } } { 3 } ) ] } \\ { = \frac { 1 } { 3 } \operatorname { Tan } ^ { - 1 } ( 1 ) - [ \frac { 1 } { 3 } \cdot \operatorname { Tan } ^ { - 1 } ( \frac { \sqrt { 3 } } { 3 } ) ] } \\ { = \frac { 1 } { 3 } ( \frac { 1 \pi } { 4 } ) - [ \frac { 1 } { 3 } ( \frac { 1 } { 6 } \pi ) ] } \\ { = \frac { 1 } { 12 } \pi - [ \frac { 1 } { 18 } \pi ] } \\ { = \frac { 1 } { 12 } \pi - \frac { 1 } { 18 } \pi = \frac { 1 } { 36 } \pi } \end{array} \right. \)
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Did you know that the integral you are solving, particularly one involving functions like \( \frac{1}{a^2 + u^2} \), is often connected to real-world applications in fields like physics and engineering? For instance, calculating areas under curves can help in understanding dynamics of systems, such as the motion of projectiles or the behavior of electrical circuits in alternating current (AC). When solving integrals like \( \int \frac{1}{a^2 + u^2} \, du \), one common mistake is neglecting to adjust the limits of integration after making a substitution. Always remember to change your limits based on the substitution you’ve made, ensuring that they reflect the values of the new variable correctly. It’s a small step, but it can save you from getting the wrong answer in the end!
