What is the integral of 10e^{2t}sin(t) ?

The integral of 10e^{2t}sin(t) is 10(-(e^{2t}cos(t))/5+(2e^{2t}sin(t))/5)+C

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integral of 10e^{2t}sin(t)

\int 10 e^{2 t} sin (t) d t

10 (- \frac{e ^{2 t} cos ( t )}{5} + \frac{2 e ^{2 t} sin ( t )}{5}) + C

Solution steps

\int 10 e^{2 t} sin (t) d t

Take the constant out:

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

= 10 \cdot \int e^{2 t} sin (t) d t

Apply Integration By Parts

: - e^{2 t} cos (t) - \int - 2 e^{2 t} cos (t) d t

= 10 (- e^{2 t} cos (t) - \int - 2 e^{2 t} cos (t) d t)

Take the constant out:

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

= 10 (- e^{2 t} cos (t) - (- 2 \cdot \int e^{2 t} cos (t) d t))

Apply Integration By Parts

: e^{2 t} sin (t) - \int e^{2 t} \cdot 2 sin (t) d t

= 10 (- e^{2 t} cos (t) - (- 2 (e^{2 t} sin (t) - \int e^{2 t} \cdot 2 sin (t) d t)))

Take the constant out:

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

= 10 (- e^{2 t} cos (t) - (- 2 (e^{2 t} sin (t) - 2 \cdot \int e^{2 t} sin (t) d t)))

Therefore

10 \cdot \int e^{2 t} sin (t) d t = 10 (- e^{2 t} cos (t) - (- 2 (e^{2 t} sin (t) - 2 \cdot \int e^{2 t} sin (t) d t)))

Isolate

\int e^{2 t} sin (t) d t

= 10 (- \frac{e ^{2 t} cos ( t )}{5} + \frac{2 e ^{2 t} sin ( t )}{5})

Add a constant to the solution

= 10 (- \frac{e ^{2 t} cos ( t )}{5} + \frac{2 e ^{2 t} sin ( t )}{5}) + C

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What is the integral of 10e^{2t}sin(t) ?
The integral of 10e^{2t}sin(t) is 10(-(e^{2t}cos(t))/5+(2e^{2t}sin(t))/5)+C