Evaluating an integral is important in order to find the area under a curve or to calculate the volume of a solid. It is also used to find the average value of a function and to solve various real-world problems in physics, engineering, and economics.
To evaluate an integral, you must first determine the appropriate integration method, such as substitution or integration by parts. Then, you must apply the fundamental theorem of calculus and use algebraic manipulation to simplify the integral into a form that can be solved using basic integration rules or tables.
The given integral can be evaluated using the substitution method. Let u = tan⁻¹x, then du = 1/(1+x²)dx. After substituting and simplifying, the integral becomes 1/2u² - 9cosx + x + C.
Yes, this integral can be solved without using a calculator as long as you are familiar with the basic integration rules and techniques. However, a calculator may be helpful in checking your answer.
The constant C, also known as the constant of integration, is added to the final answer because the derivative of a constant is always 0. This is why integrals have an infinite number of solutions, as any constant value can be added to the final answer.