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

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

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integral of e^{3x}(x^2+y^2)

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

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

Solution steps

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

Take the constant out:

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

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

Apply the Sum Rule:

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

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

\int x^{2} d y = x^{2} y

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

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

Add a constant to the solution

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

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