What is the integral from 0 to 1 of x^2e^{-13x} ?

The integral from 0 to 1 of x^2e^{-13x} is (2e^{13}-197)/(2197e^{13)}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to 1 of x^2e^{-13x}

\int_{0}^{1} x^{2} e^{- 13 x} d x

\frac{2 e ^{13} - 197}{2197 e ^{13}}

Decimal Notation

0.00091 \dots

Solution steps

\int_{0}^{1} x^{2} e^{- 13 x} d x

Apply u-substitution

: \int_{0}^{- 13} - \frac{e ^{u} u ^{2}}{2197} d u

= \int_{0}^{- 13} - \frac{e ^{u} u ^{2}}{2197} d u

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

= - \int_{- 13}^{0} - \frac{e ^{u} u ^{2}}{2197} d u

Take the constant out:

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

= - (- \frac{1}{2197} \cdot \int_{- 13}^{0} e^{u} u^{2} d u)

Apply Integration By Parts

: [u^{2} e^{u} - \int 2 u e^{u} d u]_{- 13}^{0}

= - (- \frac{1}{2197} [u^{2} e^{u} - \int 2 u e^{u} d u]_{- 13}^{0})

\int 2 u e^{u} d u = 2 (e^{u} u - e^{u})

= - (- \frac{1}{2197} [u^{2} e^{u} - 2 (e^{u} u - e^{u})]_{- 13}^{0})

Simplify

= \frac{1}{2197} [e^{u} u^{2} - 2 (e^{u} u - e^{u})]_{- 13}^{0}

Compute the boundaries:

2 - \frac{197}{e ^{13}}

= \frac{1}{2197} (2 - \frac{197}{e ^{13}})

Simplify

= \frac{2 e ^{13} - 197}{2197 e ^{13}}

\int_{0}^{1} \frac{x - 4}{x ^{2} + 5 x + 6} d x

\int_{0}^{4} x (x - 1) d x

\int_{0}^{1.2} e^{x} cos (x) d x

\int_{1}^{11} \frac{1}{x} d x

\int_{- 3}^{5} 6 x^{5} d x

What is the integral from 0 to 1 of x^2e^{-13x} ?
The integral from 0 to 1 of x^2e^{-13x} is (2e^{13}-197)/(2197e^{13)}