What is the integral of e^{t+3} ?

The integral of e^{t+3} is e^{3+t}+C

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integral of e^{t+3}

\int e^{t + 3} d t

e^{3 + t} + C

Solution steps

\int e^{t + 3} d t

e^{t + 3} = e^{3} e^{t}

= \int e^{3} e^{t} d t

Take the constant out:

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

= e^{3} \cdot \int e^{t} d t

Use the common integral:

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

= e^{3} e^{t}

Apply exponent rule:

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

e^{3} e^{t} = e^{3 + t}

= e^{3 + t}

Add a constant to the solution

= e^{3 + t} + C