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

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

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integral of e^{2x}(x+1)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

\frac{1}{2} e^{2 x} (x + 1) - \frac{1}{4} e^{2 x} : \frac{1}{2} e^{2 x} x + \frac{1}{4} e^{2 x}

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

Add a constant to the solution

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

\frac{s + 2}{s ( s ^{2} + 9 )}

\int 2 x (x^{2} + 6)^{4} d x

6 \cdot e^{5 t} \cdot cos (2 t) - e^{7 t}

\int a e^{3 x} d x

x \to - 2 lim (x^{2} + 2 x + 4)

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