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

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

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

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

2 e^{2 x} x + C

Solution steps

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

(4 x + 2) e^{2 x} = 2 (2 x + 1) e^{2 x}

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

Take the constant out:

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

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

Apply Integration By Parts

: e^{2 x} x + \frac{1}{2} e^{2 x} - \int e^{2 x} d x

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

\int e^{2 x} d x = \frac{1}{2} e^{2 x}

= 2 (e^{2 x} x + \frac{1}{2} e^{2 x} - \frac{1}{2} e^{2 x})

Add similar elements:

\frac{1}{2} e^{2 x} - \frac{1}{2} e^{2 x} = 0

= 2 e^{2 x} x

Add a constant to the solution

= 2 e^{2 x} x + C

x \to 0 lim (ln (6 x + 1))

\int \frac{x ^{3} + 16}{x ^{2} + 16} d x

x^{2} y^{^{'}} + 3 x y = \frac{sin ( 4 x )}{x}

\frac{d}{d x} (arcsin (x) d x)

f (x) = (x^{3} - 25 x)^{8}, (- 5, 0)

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