The antiderivative of (e^sin(t)) *(cos(t)) is (e^sin(t)) + C, where C is a constant.
To find the antiderivative, use the substitution technique by letting u = sin(t) and du = cos(t)dt. Then, the integral becomes ∫(e^u)du which can be easily solved using integration by parts.
No, the antiderivative of (e^sin(t)) *(cos(t)) is not an elementary function because it cannot be expressed in terms of elementary functions like polynomials, exponential functions, and trigonometric functions.
The domain of the antiderivative of (e^sin(t)) *(cos(t)) is the same as the domain of the original function, which is all real numbers.
Yes, the antiderivative of (e^sin(t)) *(cos(t)) can be used to find the definite integral by applying the fundamental theorem of calculus, which states that the definite integral of a function can be found by evaluating its antiderivative at the upper and lower limits of integration and subtracting the results.