Question

Resolve the integral from 0 to 6 of xe^x with respect to x.

Answer 1

`\int\limits^6_0 {xe^(x) } \, dx = 4e^5 +1`

Explanation:

Step(i):-

Given

`I = \int\limits^6_0 {xe^(x) } \, dx`

By using UV formula

`\int\limits {UV} \, dx = U\int\limits {v} \, dx - ( \int\limits {(U^(l) } \int\limits {v} \, dx \, )dx`

Now , we take

U = x and V = e ˣ

U¹ = 1 and
`\int\limits {e^(x) } \, dx = e^(x)`

Step(ii):-

`\int\limits {xe^x} \, dx = x\int\limits {e^x} \, dx - ( \int\limits {(1) } \int\limits {e^x} \, dx \, )dx`

= x e ˣ -
`\int\limits {e^x} \, dx`

= x e ˣ - e ˣ + C

`\int\limits^6_0 {xe^(x) } \, dx =( e^ x ( x -1) )_(0) ^(6)`

= e⁵ (5-1) - (e⁰ (0-1)

= 4 e⁵ + 1

Final answer:-

`\int\limits^6_0 {xe^(x) } \, dx = 4e^5 +1`