What is the integral of-7xe^{x^2} ?

The integral of-7xe^{x^2} is -7/2 e^{x^2}+C

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integral of-7xe^{x^2}

\int - 7 x e^{x^{2}} d x

- \frac{7}{2} e^{x^{2}} + C

Solution steps

\int - 7 x e^{x^{2}} d x

Take the constant out:

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

= - 7 \cdot \int x e^{x^{2}} d x

Apply u-substitution

: \int \frac{e ^{u}}{2} d u

= - 7 \cdot \int \frac{e ^{u}}{2} d u

Take the constant out:

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

= - 7 \cdot \frac{1}{2} \cdot \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= - 7 \cdot \frac{1}{2} e^{u}

Substitute back

u = x^{2}

= - 7 \cdot \frac{1}{2} e^{x^{2}}

Simplify

- 7 \cdot \frac{1}{2} e^{x^{2}} : - \frac{7}{2} e^{x^{2}}

= - \frac{7}{2} e^{x^{2}}

Add a constant to the solution

= - \frac{7}{2} e^{x^{2}} + C

\int x^{2} 2 x^{3} + 1 d x

2 y^{^{'}} + 4 y = - 8 y^{3}

y = f (y, x) (g (x))

\frac{d}{d x} (x y z)