Use integration by parts to find the given integral. Use symbolic notation and fractions where needed. Integra sign has 1 on top and 0 on bottom....2x(e^(-3x)  + e^(-x)) dx

Expert Answers

An illustration of the letter 'A' in a speech bubbles

You need to evaluate the definite integral, such that:

`int_0^1 2x(e^(-3x) + e^(-x))dx`

You need to open the brackets such that:

`int_0^1 (2xe^(-3x) + 2xe^(-x))dx`

You need to use the property of linearity of integral, such that:

`int_0^1 (2xe^(-3x) + 2xe^(-x))dx = int_0^1 (2xe^(-3x))dx + int_0^1 (2xe^(-x))dx`

You need to solve the integrals, using integration by parts, such that:

`int_a^b udv = uv|_a^b - int_a^b vdu`

Considering `u = x` and `dv = e^(-3x)dx` yields:

`u = x => du = dx`

`dv = e^(-3x)dx => v = -(e^(-3x))/3`

`int_0^1 (2xe^(-3x))dx = 2int_0^1 (xe^(-3x))dx`

`int_0^1 (xe^(-3x))dx = -x(e^(-3x))/3|_0^1 + (1/3)int_0^1 e^(-3x)dx`

`int_0^1 (xe^(-3x))dx = -x(e^(-3x))/3|_0^1 - (1/9)e^(-3x)|_0^1`

`2int_0^1 (xe^(-3x))dx = 2(-x(e^(-3x))/3|_0^1 - (1/9)e^(-3x)|_0^1)`

`2int_0^1 (xe^(-3x))dx = 2(-1*e^(-3)/3 - e^(-3)/9 + e^0/9)`

`2int_0^1 (xe^(-3x))dx = -2/(3e^3) - 1/(9e^3) + 1/9`

`2int_0^1 (xe^(-3x))dx = -7/(9e^3) + 1/9`

`2int_0^1 (xe^(-3x))dx = (e^3 - 7)/(9e^3)`

Solving the integral `int_0^1 (2xe^(-x))dx` using parts yields:

`u = x => du = dx`

`dv = e^(-x)dx => v = -e^(-x)`

`int_0^1 (2xe^(-x))dx = 2int_0^1 (xe^(-x))dx `

`2int_0^1 (xe^(-x))dx = 2(-xe^(-x)|_0^1 - e^(-x)|_0^1)`

`2int_0^1 (xe^(-x))dx = 2(-1/e - 1/e + e^0) 2int_0^1 (xe^(-x))dx = -4/e + 1 => 2int_0^1 (xe^(-x))dx = (e - 4)/e`

`int_0^1 (2xe^(-3x) + 2xe^(-x))dx = (e^3 - 7)/(9e^3) +(e - 4)/e`

Hence, evaluating the given definite integral using parts, yields` int_0^1 (2xe^(-3x) + 2xe^(-x))dx = (e^3 - 7)/(9e^3) +(e - 4)/e.`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial