What is the integral of e^{x^2+y^2}sin(y) ?

The integral of e^{x^2+y^2}sin(y) is sin(y)e^{y^2}(sqrt(pi))/2 \erfi(x)+C

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integral of e^{x^2+y^2}sin(y)

\int e^{x^{2} + y^{2}} sin (y) d x

sin (y) e^{y^{2}} \frac{π}{2} erfi (x) + C

Solution steps

\int e^{x^{2} + y^{2}} sin (y) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= sin (y) \cdot \int e^{y^{2} + x^{2}} d x

Simplify

e^{y^{2} + x^{2}} : e^{y^{2}} e^{x^{2}}

= sin (y) \cdot \int e^{y^{2}} e^{x^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= sin (y) e^{y^{2}} \cdot \int e^{x^{2}} d x

Use the nonelementary integral:

\int e^{x^{2}} d x = \frac{π}{2} erfi (x)

= sin (y) e^{y^{2}} \frac{π}{2} erfi (x)

Add a constant to the solution

= sin (y) e^{y^{2}} \frac{π}{2} erfi (x) + C

What is the integral of e^{x^2+y^2}sin(y) ?
The integral of e^{x^2+y^2}sin(y) is sin(y)e^{y^2}(sqrt(pi))/2 \erfi(x)+C