The purpose of integrating dx/(x*sqrt(9+16x^2)) is to find the antiderivative or the original function from which this expression is derived. This process is useful in solving real-world problems in various fields of science and engineering.
The first step is to use the substitution method by letting u = 9+16x^2 and then finding the value of du. Next, rewrite the expression as dx/(x*sqrt(u)) and use the power rule for integration to solve for the antiderivative. Finally, substitute back the original variable x to get the final answer.
Yes, there are other methods such as using trigonometric substitution, partial fractions, or integration by parts. However, the substitution method is the most efficient and straightforward way to solve this particular expression.
Some common mistakes to avoid when integrating dx/(x*sqrt(9+16x^2)) include not using the correct substitution, making errors in algebraic manipulation, and forgetting to include the constant of integration. It is important to carefully follow the steps and double-check the final answer.
The solution to dx/(x*sqrt(9+16x^2)) can be used to solve problems in physics, engineering, and other fields that involve calculating the area under a curve. It can also be applied in determining the velocity of an object, the amount of work done, or the distance traveled.