How do you integrate #int e^x sin sqrtx dx # using integration by parts?

Question

How do you integrate #int e^x  sin sqrtx   dx  # using integration by parts?

Evelyn Agard · Accepted Answer

I got overenthusiastic but I got stuck....I am not sure about it...I suspect it is either very complicated or not possible directly...

I got stuck...

Isaiah Allen · Answer

You do not.

You are not going to find a satisfactory answer to this integral. That is, the result cannot be represented by elementary functions. For reference, an acceptable result of this integral would be: #int (e^xsin(sqrt(x)))dx = -(1/4)i(root4(e) sqrt(pi) " erf"(1/2 - isqrt(x)) - root4(e)sqrt(pi) " erf"(1/2 + isqrt(x)) + 2e^(x-isqrt(x))(-1+e^(2isqrt(x)))) + C# where #"erf"(x) = 2/(sqrt(pi)) int_0^x e^(-t^2)dt# and #i# is the imaginary number. You would not encounter this type of integral in a high school or college level calculus class. In fact, you would not see an integral of this type even while pursuing an undergraduate mathematics degree.

Charles Anctil · Answer

undefined To integrate $ \int e^x \sin(\sqrt{x}) \, dx $ using integration by parts, we follow these steps:

1. Choose $ u $ and $ dv $:
   Let $ u = \sin(\sqrt{x}) $ (trigonometric function) and $ dv = e^x \, dx $ (exponential function).

2. Compute the differentials:
   Calculate the differentials:
   $ du = \frac{1}{2\sqrt{x}} \cos(\sqrt{x}) \, dx $ and $ v = \int e^x \, dx = e^x $.

3. Apply the integration by parts formula:
   The integration by parts formula is given as:
   $ \int u \, dv = uv - \int v \, du $.
   Substitute the values into the formula:
   $ \int e^x \sin(\sqrt{x}) \, dx = -e^x \sin(\sqrt{x}) + \int \frac{1}{2\sqrt{x}} e^x \cos(\sqrt{x}) \, dx $.

4. Integrate the remaining integral:
   The integral $ \int \frac{1}{2\sqrt{x}} e^x \cos(\sqrt{x}) \, dx $ can be integrated using integration by parts again or through other techniques such as substitution or further simplification.

Thus, by using integration by parts, we've expressed the original integral in terms of another integral that may need further techniques for evaluation.