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

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

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

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

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

Solution steps

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

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

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

Take the constant out:

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

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

Simplify

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

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 s + 2 s^{2}

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

Simplify

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

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

Add a constant to the solution

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

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