What is the integral from 0 to 6 of 7000te^{-0.1t} ?

The integral from 0 to 6 of 7000te^{-0.1t} is 85330.96757…

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integral from 0 to 6 of 7000te^{-0.1t}

\int_{0}^{6} 7000 t e^{- 0.1 t} d t

85330.96757 \dots

Solution steps

\int_{0}^{6} 7000 t e^{- 0.1 t} d t

Take the constant out:

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

= 7000 \cdot \int_{0}^{6} t e^{- 0.1 t} d t

Apply u-substitution

: \int_{0}^{- 0.6} \frac{10}{0.1} e^{u} u d u

= 7000 \cdot \int_{0}^{- 0.6} \frac{10}{0.1} e^{u} u d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 7000 (- \int_{- 0.6}^{0} \frac{10}{0.1} e^{u} u d u)

Take the constant out:

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

= 7000 (- \frac{10}{0.1} \cdot \int_{- 0.6}^{0} e^{u} u d u)

Simplify

= 7000 (- 100 \cdot \int_{- 0.6}^{0} e^{u} u d u)

Apply Integration By Parts

: [e^{u} u - \int e^{u} d u]_{- 0.6}^{0}

= 7000 (- 100 [e^{u} u - \int e^{u} d u]_{- 0.6}^{0})

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

= 7000 (- 100 [e^{u} u - e^{u}]_{- 0.6}^{0})

Simplify

= - 700000 [e^{u} u - e^{u}]_{- 0.6}^{0}

Compute the boundaries:

- 0.12190 \dots

= - 700000 (- 0.12190 \dots)

Simplify

= 85330.96757 \dots

\int_{1}^{10} \frac{5}{x ^{3}} d x

\int \frac{6 x ^{2} + 7 x + 6}{( x ^{2} + 1 ) ^{2}} d x

(- 2, 10), (4, - 2)

\frac{d y}{d x} = x^{2} - 1

\frac{d}{d x} (ln (2 x) - \frac{1}{4} x)

What is the integral from 0 to 6 of 7000te^{-0.1t} ?
The integral from 0 to 6 of 7000te^{-0.1t} is 85330.96757…