What is the integral of-2e^{x^2}sin(2y)+(x^2)/2+1 ?

The integral of-2e^{x^2}sin(2y)+(x^2)/2+1 is x+(x^3)/6-sqrt(pi)sin(2y)\erfi(x)+C

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integral of-2e^{x^2}sin(2y)+(x^2)/2+1

\int - 2 e^{x^{2}} sin (2 y) + \frac{x ^{2}}{2} + 1

x + \frac{x ^{3}}{6} - π sin (2 y) erfi (x) + C

Solution steps

\int - 2 e^{x^{2}} sin (2 y) + \frac{x ^{2}}{2} + 1 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d x + \int \frac{x ^{2}}{2} d x - \int 2 sin (2 y) e^{x^{2}} d x

\int 1 d x = x

\int \frac{x ^{2}}{2} d x = \frac{x ^{3}}{6}

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

= x + \frac{x ^{3}}{6} - π sin (2 y) erfi (x)

Add a constant to the solution

= x + \frac{x ^{3}}{6} - π sin (2 y) erfi (x) + C

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What is the integral of-2e^{x^2}sin(2y)+(x^2)/2+1 ?
The integral of-2e^{x^2}sin(2y)+(x^2)/2+1 is x+(x^3)/6-sqrt(pi)sin(2y)\erfi(x)+C