What is the integral of (2+2x)e^{4x+2x^2} ?

The integral of (2+2x)e^{4x+2x^2} is 1/2 e^{4x+2x^2}+C

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integral of (2+2x)e^{4x+2x^2}

\int (2 + 2 x) e^{4 x + 2 x^{2}} d x

\frac{1}{2} e^{4 x + 2 x^{2}} + C

Solution steps

\int (2 + 2 x) e^{4 x + 2 x^{2}} d x

2 + 2 x = 2 (1 + \frac{2 x}{2})

= \int 2 (1 + \frac{2 x}{2}) e^{4 x + 2 x^{2}} d x

Take the constant out:

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

= 2 \cdot \int (1 + \frac{2 x}{2}) e^{4 x + 2 x^{2}} d x

Simplify

= 2 \cdot \int e^{4 x + 2 x^{2}} (1 + x) d x

Apply u-substitution

= 2 \cdot \int \frac{e ^{u}}{4} d u

Take the constant out:

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

= 2 \cdot \frac{1}{4} \cdot \int e^{u} d u

Use the common integral:

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

= 2 \cdot \frac{1}{4} e^{u}

Substitute back

u = 4 x + 2 x^{2}

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

Simplify

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

= \frac{1}{2} e^{4 x + 2 x^{2}}

Add a constant to the solution

= \frac{1}{2} e^{4 x + 2 x^{2}} + C

What is the integral of (2+2x)e^{4x+2x^2} ?
The integral of (2+2x)e^{4x+2x^2} is 1/2 e^{4x+2x^2}+C