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

The integral of e^{y+x} is e^{x+y}+C

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integral of e^{y+x}

\int e^{y + x} d y

e^{x + y} + C

Solution steps

\int e^{y + x} d y

e^{x + y} = e^{x} e^{y}

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

Take the constant out:

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

= e^{x} \cdot \int e^{y} d y

Use the common integral:

\int e^{y} d y = e^{y}

= e^{x} e^{y}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{x} e^{y} = e^{x + y}

= e^{x + y}

Add a constant to the solution

= e^{x + y} + C