What is the integral of 6xe^{x^2}sin(e^{x^2}) ?

The integral of 6xe^{x^2}sin(e^{x^2}) is -3cos(e^{x^2})+C

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integral of 6xe^{x^2}sin(e^{x^2})

\int 6 x e^{x^{2}} sin (e^{x^{2}}) d x

- 3 cos (e^{x^{2}}) + C

Solution steps

\int 6 x e^{x^{2}} sin (e^{x^{2}}) d x

Take the constant out:

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

= 6 \cdot \int x e^{x^{2}} sin (e^{x^{2}}) d x

Apply u-substitution

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

= 6 \cdot \int \frac{e ^{u} sin ( e ^{u} )}{2} d u

Take the constant out:

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

= 6 \cdot \frac{1}{2} \cdot \int e^{u} sin (e^{u}) d u

Apply u-substitution

: \int sin (v) d v

= 6 \cdot \frac{1}{2} \cdot \int sin (v) d v

Use the common integral:

\int sin (v) d v = - cos (v)

= 6 \cdot \frac{1}{2} (- cos (v))

Substitute back

= 6 \cdot \frac{1}{2} (- cos (e^{x^{2}}))

Simplify

6 \cdot \frac{1}{2} (- cos (e^{x^{2}})) : - 3 cos (e^{x^{2}})

= - 3 cos (e^{x^{2}})

Add a constant to the solution

= - 3 cos (e^{x^{2}}) + C

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What is the integral of 6xe^{x^2}sin(e^{x^2}) ?
The integral of 6xe^{x^2}sin(e^{x^2}) is -3cos(e^{x^2})+C