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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{e ^{u} sin ( a u )}{2} d u

= \int \frac{e ^{u} sin ( a u )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int e^{u} sin (a u) d u

Apply Integration By Parts

: - e^{u} \frac{1}{a} cos (a u) - \int - e^{u} \frac{1}{a} cos (a u) d u

= \frac{1}{2} (- e^{u} \frac{1}{a} cos (a u) - \int - e^{u} \frac{1}{a} cos (a u) d u)

Take the constant out:

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

= \frac{1}{2} (- e^{u} \frac{1}{a} cos (a u) - (- \frac{1}{a} \cdot \int e^{u} cos (a u) d u))

Apply Integration By Parts

: e^{u} \frac{1}{a} sin (a u) - \int e^{u} \frac{1}{a} sin (a u) d u

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

Take the constant out:

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

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

Therefore

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

Isolate

\int e^{u} sin (a u) d u

= \frac{1}{2} (- \frac{a e ^{u} cos ( a u )}{a ^{2} + 1} + \frac{e ^{u} sin ( a u )}{a ^{2} + 1})

Substitute back

u = x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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