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

The integral of ()(e^{2t}x) is d/(dt)(xe^{2t})t-x(2e^{2t}t-e^{2t})+C

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integral of ()(e^{2t}x)

\int \frac{d}{d t} (e^{2 t} x) d t

\frac{d}{d t} (x e^{2 t}) t - x (2 e^{2 t} t - e^{2 t}) + C

Solution steps

\int \frac{d}{d t} (e^{2 t} x) d t

Apply Integration By Parts

: \frac{d}{d t} (x e^{2 t}) t - \int 4 x e^{2 t} t d t

= \frac{d}{d t} (x e^{2 t}) t - \int 4 x e^{2 t} t d t

\int 4 x e^{2 t} t d t = x (2 e^{2 t} t - e^{2 t})

= \frac{d}{d t} (x e^{2 t}) t - x (2 e^{2 t} t - e^{2 t})

Add a constant to the solution

= \frac{d}{d t} (x e^{2 t}) t - x (2 e^{2 t} t - e^{2 t}) + C

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What is the integral of ()(e^{2t}x) ?
The integral of ()(e^{2t}x) is d/(dt)(xe^{2t})t-x(2e^{2t}t-e^{2t})+C