Painguy said:Homework Statement
∫ 3+ x√x from -1 to 4
Homework Equations
The Attempt at a Solution
∫ 3+ x√x from -1 to 4 = 3x+(2(x^5/2))/5 evaluated from -1 to 4
(12 + (2(4^5/2))/5) +(3 +(2(-1)^5/2)/5) ?
15+64/5 +(2(-1^(5/2))/5)
139/5 -(2(-1^(5/2)))/5
139/5 -2i/5
is that right?
Because of the term with the square root, the integrand is defined only for x ≥ 0. What is the complete statement of the problem?Painguy said:Homework Statement
∫ 3+ x√x from -1 to 4
Painguy said:Homework Equations
The Attempt at a Solution
∫ 3+ x√x from -1 to 4 = 3x+(2(x^5/2))/5 evaluated from -1 to 4
(12 + (2(4^5/2))/5) +(3 +(2(-1)^5/2)/5) ?
15+64/5 +(2(-1^(5/2))/5)
139/5 -(2(-1^(5/2)))/5
139/5 -2i/5
is that right?
A definite integral with complex result is an integration problem where the integral evaluates to a complex number. This means that the result of the integral will have a real and imaginary component.
A regular definite integral evaluates to a real number, while a definite integral with complex result evaluates to a complex number. This is because the function being integrated may have complex values, or the limits of integration may be complex numbers.
Functions that have complex values, such as trigonometric functions like sine and cosine, can result in a definite integral with complex result. Additionally, functions with complex exponents, such as e^z, can also lead to a complex result.
A definite integral with complex result can be useful in solving complex physics and engineering problems, as well as in signal processing and control systems. It can also be used in probability and statistics to calculate complex probabilities.
To evaluate a definite integral with complex result, one can use the same techniques as for regular definite integrals, such as the fundamental theorem of calculus, substitution, and integration by parts. However, it is important to keep track of the real and imaginary components separately and use complex arithmetic when necessary.