What is the integral of e^{(1-s)t} ?

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

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integral of e^{(1-s)t}

\int e^{(1 - s) t} d t

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

Solution steps

\int e^{(1 - s) t} d t

Apply Integration By Parts

: e^{(1 - s) t} t - \int e^{(1 - s) t} (1 - s) t d t

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

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

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

Add a constant to the solution

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

(4 + x) y^{^{'}} = 9 y

\frac{\partial}{\partial x} (y^{2} - 4 x)

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

\int \frac{x ^{2} - 4}{4 x} d x

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