What is the integral of e^{3x-2y}+x^2e^{-2y} ?

The integral of e^{3x-2y}+x^2e^{-2y} is e^{-2y}(1/3 e^{3x}+(x^3)/3)+C

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integral of e^{3x-2y}+x^2e^{-2y}

\int e^{3 x - 2 y} + x^{2} e^{- 2 y} d x

e^{- 2 y} (\frac{1}{3} e^{3 x} + \frac{x ^{3}}{3}) + C

Solution steps

\int e^{3 x - 2 y} + x^{2} e^{- 2 y} d x

e^{- 2 y + 3 x} + e^{- 2 y} x^{2} = e^{- 2 y} (e^{3 x} + x^{2})

= \int e^{- 2 y} (e^{3 x} + x^{2}) d x

Take the constant out:

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

= e^{- 2 y} \cdot \int e^{3 x} + x^{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= e^{- 2 y} (\int e^{3 x} d x + \int x^{2} d x)

\int e^{3 x} d x = \frac{1}{3} e^{3 x}

\int x^{2} d x = \frac{x ^{3}}{3}

= e^{- 2 y} (\frac{1}{3} e^{3 x} + \frac{x ^{3}}{3})

Add a constant to the solution

= e^{- 2 y} (\frac{1}{3} e^{3 x} + \frac{x ^{3}}{3}) + C

\int \frac{( sin ( x ) + 1 ) ^{\frac{3}{2}}}{sec ( x )} d x

\int 1 + (\frac{3 x}{2})^{2} d x

f (x) = 6 x^{2} + 2 x - 6

\frac{d}{d x} (\frac{x + 1}{x - 3})

y = arcsin (6 x + 1)

What is the integral of e^{3x-2y}+x^2e^{-2y} ?
The integral of e^{3x-2y}+x^2e^{-2y} is e^{-2y}(1/3 e^{3x}+(x^3)/3)+C