What is the integral of xe^{x^2}sin(x^2) ?

The integral of xe^{x^2}sin(x^2) is 1/4 e^{x^2}sin(x^2)-1/4 e^{x^2}cos(x^2)+C

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integral of xe^{x^2}sin(x^2)

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

\frac{1}{4} e^{x^{2}} sin (x^{2}) - \frac{1}{4} e^{x^{2}} cos (x^{2}) + C

Solution steps

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

Apply Integration By Parts

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

\int e^{x^{2}} x cos (x^{2}) d x = \frac{1}{4} (e^{x^{2}} sin (x^{2}) + e^{x^{2}} cos (x^{2}))

= \frac{1}{2} e^{x^{2}} sin (x^{2}) - \frac{1}{4} (e^{x^{2}} sin (x^{2}) + e^{x^{2}} cos (x^{2}))

Simplify

\frac{1}{2} e^{x^{2}} sin (x^{2}) - \frac{1}{4} (e^{x^{2}} sin (x^{2}) + e^{x^{2}} cos (x^{2})) : \frac{1}{4} e^{x^{2}} sin (x^{2}) - \frac{1}{4} e^{x^{2}} cos (x^{2})

= \frac{1}{4} e^{x^{2}} sin (x^{2}) - \frac{1}{4} e^{x^{2}} cos (x^{2})

Add a constant to the solution

= \frac{1}{4} e^{x^{2}} sin (x^{2}) - \frac{1}{4} e^{x^{2}} cos (x^{2}) + C

What is the integral of xe^{x^2}sin(x^2) ?
The integral of xe^{x^2}sin(x^2) is 1/4 e^{x^2}sin(x^2)-1/4 e^{x^2}cos(x^2)+C