What is the integral from-1 to ln(2) of (3t)/(e^t) ?

The integral from-1 to ln(2) of (3t)/(e^t) is (3(-ln(2)-1))/2

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integral from-1 to ln(2) of (3t)/(e^t)

\int_{- 1}^{l n (2)} \frac{3 t}{e ^{t}} d t

\frac{3 ( - ln ( 2 ) - 1 )}{2}

Decimal

- 2.53972 \dots

Solution steps

\int_{- 1}^{l n (2)} \frac{3 t}{e ^{t}} d t

Take the constant out:

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

= 3 \cdot \int_{- 1}^{l n (2)} \frac{t}{e ^{t}} d t

Apply Integration By Parts

: [\frac{- t - e ^{t} \cdot \int - \frac{1}{e ^{t}} d t}{e ^{t}}]_{- 1}^{l n (2)}

= 3 [\frac{- t - e ^{t} \cdot \int - \frac{1}{e ^{t}} d t}{e ^{t}}]_{- 1}^{l n (2)}

\int - \frac{1}{e ^{t}} d t = \frac{1}{e ^{t}}

= 3 [\frac{- t - e ^{t} \frac{1}{e ^{t}}}{e ^{t}}]_{- 1}^{l n (2)}

e^{t} \frac{1}{e ^{t}} = 1

= 3 [\frac{- t - 1}{e ^{t}}]_{- 1}^{l n (2)}

Compute the boundaries:

\frac{- ln ( 2 ) - 1}{2}

= 3 \cdot \frac{- ln ( 2 ) - 1}{2}

Simplify

= \frac{3 ( - ln ( 2 ) - 1 )}{2}

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What is the integral from-1 to ln(2) of (3t)/(e^t) ?
The integral from-1 to ln(2) of (3t)/(e^t) is (3(-ln(2)-1))/2