The process for evaluating this integral involves using substitution to simplify the expression. First, let u = x+3, then du = dx. The integral then becomes INT sqrt(u^2-9) du. Using the substitution u = 3sec(theta), the expression can be simplified further to become INT 3sec(theta) tan(theta) d(theta). This can then be solved using the trigonometric identity cos^2(theta) + sin^2(theta) = 1.
Substitution is often used in integrals that involve a polynomial under a square root. In this case, the expression x^2+6x can be simplified using the substitution u = x+3. If the integral involves a polynomial under a square root, it is a good indication that substitution may be useful.
Yes, there is another method for evaluating this integral without using substitution. You can use integration by parts, where the integral is split into two parts and each part is integrated separately. This method can also be used to solve the integral INT sqrt(x^2+6x) dx.
Yes, you can use a calculator to evaluate this integral. However, the calculator may not show the steps involved in solving the integral. It is always recommended to show your work and understand the steps involved in solving the integral.
Yes, there are many real-life applications of this integral. It can be used to calculate the length of a curve, the area under a curve, and the volume of a solid of revolution. This integral is also commonly used in physics and engineering to solve problems involving velocity, acceleration, and distance.