Integral Calculus Question?

Question

Is there a general formula for the antiderivative of e^[(f(x)], such that f(x) includes trigononmetric, hyperbolic, inverse trigonometric, reciprocal, and power functions? If not, can anyone find the antiderivative of e^(3/x)?

Answer 1

There is no general formula because there exist no general elementary antiderivative for e^{f(x)}.
In general you can expect only a numerical solution of such an integral.

Nevertheless the solution of such integrals is commonly expressed in terms of (incomplete) gamma functions. You can transform such an integral by an appropriate substitution to an integral or a set of integrals of the type:
∫ t^(a-1) · e^(-t) dt
which is the gamma integral. The gamma function and the incomplete gamma functions are defined by the following definite integrals:
gamma function:
Γ(a) = ∫ t^(a-1) · e^(-t) dt from 0 to infinity
upper incomplete gamma function:
Γ(a,b) = ∫ t^(a-1) · e^(-t) dt from b to infinity
lower incomplete gamma:
γ(a,b) = ∫ t^(a-1) · e^(-t) dt from 0 to a

The values of the functions can be taken from data tables or evaluated numerically.

For your function
f(x) = 3/x
substitute t = -3/x → dx = 3/t² dt
Hence:
∫ e^(3/x) dx
= ∫ e^(-t) · 3/t² dt
= 3 · ∫ t^(-2) · e^(-t) dt
= 3 · γ(-1,t) + c
= 3 · γ(-1, -3/x) + c

Answer 2

No. In fact, the vast majority of such functions have no elementary antiderivative (meaning that the antiderivative cannot be expressed at all in terms of rational, exponential, logarithmic, and rational functions). e^(3/x) is, I'm sorry to say, one of the functions that can't be integrated in closed form.

Answer 3

Integration is opposite technique to differentiation. in case you're conversing approximately indefinite indispensable it has lot many formulae to undergo in suggestions. For sure indispensable its in elementary terms arithmetical calculations. the undertaking point relies upon on what you would be studying and wherein area of the worldwide. in case you're making the question extra informative, it is going to likely be better. yet asking your seniors approximately it is going to likely be a sturdy thought.