What is the integral of 6xe^{3x^2} ?

The integral of 6xe^{3x^2} is e^{3x^2}+C

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integral of 6xe^{3x^2}

\int 6 x e^{3 x^{2}} d x

e^{3 x^{2}} + C

Solution steps

\int 6 x e^{3 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^{3 x^{2}} d x

Apply u-substitution

: \int \frac{e ^{u}}{6} d u

= 6 \cdot \int \frac{e ^{u}}{6} d u

Take the constant out:

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

= 6 \cdot \frac{1}{6} \cdot \int e^{u} d u

Use the common integral:

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

= 6 \cdot \frac{1}{6} e^{u}

Substitute back

u = 3 x^{2}

= 6 \cdot \frac{1}{6} e^{3 x^{2}}

Simplify

6 \cdot \frac{1}{6} e^{3 x^{2}} : e^{3 x^{2}}

= e^{3 x^{2}}

Add a constant to the solution

= e^{3 x^{2}} + C

a y^{^{''}} + b y^{^{'}} = 0

\frac{d y}{d x} + \frac{2 y}{x} = 2 x y^{\frac{3}{2}}

\int_{1}^{2} x - 1 d x

f (x) = 6 6 x^{10} + 7 x^{6}

(0, - 10), (5, - 6)