What is the integral of e^{(3-s)t} ?

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

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integral of e^{(3-s)t}

\int e^{(3 - s) t} d t

e^{(3 - s) t} t - (3 - s) \frac{1}{( 3 - s ) ^{2}} (e^{(3 - s) t} (3 - s) t - e^{(3 - s) t}) + C

Solution steps

\int e^{(3 - s) t} d t

Apply Integration By Parts

: e^{(3 - s) t} t - \int e^{(3 - s) t} (3 - s) t d t

= e^{(3 - s) t} t - \int e^{(3 - s) t} (3 - s) t d t

\int e^{(3 - s) t} (3 - s) t d t = (3 - s) \frac{1}{( 3 - s ) ^{2}} (e^{(3 - s) t} (3 - s) t - e^{(3 - s) t})

= e^{(3 - s) t} t - (3 - s) \frac{1}{( 3 - s ) ^{2}} (e^{(3 - s) t} (3 - s) t - e^{(3 - s) t})

Add a constant to the solution

= e^{(3 - s) t} t - (3 - s) \frac{1}{( 3 - s ) ^{2}} (e^{(3 - s) t} (3 - s) t - e^{(3 - s) t}) + C

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