What is the integral of e^{4x^2}8x ?

The integral of e^{4x^2}8x is e^{4x^2}+C

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integral of e^{4x^2}8x

\int e^{4 x^{2}} 8 x d x

e^{4 x^{2}} + C

Solution steps

\int e^{4 x^{2}} \cdot 8 x d x

Take the constant out:

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

= 8 \cdot \int e^{4 x^{2}} x d x

Apply u-substitution

: \int \frac{e ^{u}}{8} d u

= 8 \cdot \int \frac{e ^{u}}{8} d u

Take the constant out:

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

= 8 \cdot \frac{1}{8} \cdot \int e^{u} d u

Use the common integral:

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

= 8 \cdot \frac{1}{8} e^{u}

Substitute back

u = 4 x^{2}

= 8 \cdot \frac{1}{8} e^{4 x^{2}}

Simplify

8 \cdot \frac{1}{8} e^{4 x^{2}} : e^{4 x^{2}}

= e^{4 x^{2}}

Add a constant to the solution

= e^{4 x^{2}} + C

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