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

The integral of e^{2x}sin^2(x) is sin^2(x) 1/2 e^{2x}-1/8 (-e^{2x}cos(2x)+e^{2x}sin(2x))+C

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

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

sin^{2} (x) \frac{1}{2} e^{2 x} - \frac{1}{8} (- e^{2 x} cos (2 x) + e^{2 x} sin (2 x)) + C

Solution steps

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

Apply Integration By Parts

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

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

\int sin (2 x) \frac{1}{2} e^{2 x} d x = \frac{1}{8} (- e^{2 x} cos (2 x) + e^{2 x} sin (2 x))

= sin^{2} (x) \frac{1}{2} e^{2 x} - \frac{1}{8} (- e^{2 x} cos (2 x) + e^{2 x} sin (2 x))

Add a constant to the solution

= sin^{2} (x) \frac{1}{2} e^{2 x} - \frac{1}{8} (- e^{2 x} cos (2 x) + e^{2 x} sin (2 x)) + C

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What is the integral of e^{2x}sin^2(x) ?
The integral of e^{2x}sin^2(x) is sin^2(x) 1/2 e^{2x}-1/8 (-e^{2x}cos(2x)+e^{2x}sin(2x))+C