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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

= - e^{- x} cos (x) - \int e^{- x} cos (x) d x

Apply Integration By Parts

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

= - e^{- x} cos (x) - (e^{- x} sin (x) - \int - e^{- x} sin (x) d x)

Take the constant out:

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

= - e^{- x} cos (x) - (e^{- x} sin (x) - (- \int e^{- x} sin (x) d x))

Apply Integration By Parts

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

= - e^{- x} cos (x) - (e^{- x} sin (x) - (- (- e^{- x} cos (x) - \int e^{- x} cos (x) d x)))

Therefore

- \int e^{- x} cos (x) d x - cos (x) e^{- x} = - e^{- x} cos (x) - (e^{- x} sin (x) - (- (- e^{- x} cos (x) - \int e^{- x} cos (x) d x)))

Isolate

\int e^{- x} cos (x) d x

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

Simplify

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

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

Add a constant to the solution

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

y^{^{''}} + 4 y^{^{'}} + 4 y = x^{2} + e^{- 2 x}

\int_{0}^{4} \frac{1}{( x - 4 ) ^{2}} d x

\int \frac{4 - ln ( x + 3 )}{x + 3} d x

\frac{d}{d y} (4 x y)

x \to - 2 lim (- 4 x^{3} + 4 x + 5)

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