What is the integral from 0 to x of (1-t)/(e^t) ?

The integral from 0 to x of (1-t)/(e^t) is x/(e^x)

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integral from 0 to x of (1-t)/(e^t)

\int_{0}^{x} \frac{1 - t}{e ^{t}} d t

\frac{x}{e ^{x}}

Solution steps

\int_{0}^{x} \frac{1 - t}{e ^{t}} d t

Apply u-substitution

: \int_{1}^{e^{x}} \frac{1 - ln ( u )}{u ^{2}} d u

= \int_{1}^{e^{x}} \frac{1 - ln ( u )}{u ^{2}} d u

Apply Integration By Parts

: [- \frac{1 - ln ( u )}{u} - \int \frac{1}{u ^{2}} d u]_{1}^{e^{x}}

= [- \frac{1 - ln ( u )}{u} - \int \frac{1}{u ^{2}} d u]_{1}^{e^{x}}

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

= [- \frac{1 - ln ( u )}{u} - (- \frac{1}{u})]_{1}^{e^{x}}

Simplify

[- \frac{1 - ln ( u )}{u} - (- \frac{1}{u})]_{1}^{e^{x}} : [\frac{ln ( u )}{u}]_{1}^{e^{x}}

= [\frac{ln ( u )}{u}]_{1}^{e^{x}}

Compute the boundaries:

\frac{x}{e ^{x}}

= \frac{x}{e ^{x}}

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What is the integral from 0 to x of (1-t)/(e^t) ?
The integral from 0 to x of (1-t)/(e^t) is x/(e^x)