Hypatio said:Homework Statement
Homework Equations
solve the definite integral
[tex]\int_{2.6}^{5.5} \frac{1}{x^2+9}dx[/tex]
The Attempt at a Solution
ln(5.5^2+9)-ln(2.6^2+9) doesn't seem correct
A definite integral with x^2+c in the denominator is a mathematical expression that represents the area under a curve, where the curve is defined by the function f(x) = x^2+c. This type of integral is also known as a rational function integral.
To solve a definite integral with x^2+c in the denominator, you can use the substitution method or the partial fractions method. The substitution method involves substituting a new variable for x, while the partial fractions method involves breaking down the rational function into simpler fractions.
Definite integrals with x^2+c in the denominator are important in mathematics because they allow us to calculate the area under a curve, which has many real-world applications. They are also used in finding the volume of irregular shapes and in solving differential equations.
Yes, there are special cases when solving a definite integral with x^2+c in the denominator. One such case is when c=0, in which case the integral simplifies to the basic function f(x) = x^2. Another special case is when c is a negative number, which can lead to imaginary solutions.
Definite integrals with x^2+c in the denominator have various real-life applications. For example, they can be used to calculate the amount of fluid in a container with a curved bottom, or to find the work done by a varying force. They are also used in physics to determine the displacement of an object under a varying acceleration.