Question

Evaluate ∫4x cos(2-3x)dx

Answer 1

`{ \sf{ \int 4x \cos(2 - 3x) dx = {4x}^(2) \cos(2 - 3x) + c }}`

Explanation:

From integration by parts:

`{ \boxed{ \bf{ \int u \: (dv)/(dx) = uv - \int v \: (du)/(dx) }}}`

let u be cos(2-3x) and let dv/dx be 4x:

`{ \sf{ (du)/(dx) = 3 \sin(2 - 3x) }} \\ \\ { \sf{ v = \int 4x = 2 {x}^(2) }}`

Substitute in formular box:

`{ \sf{ = (2 {x}^(2) * \cos(2 - 3x)) - \int 2 {x}^(2) .3 \sin(2 - 3x) dx}} \\ = { \sf{2 {x}^(2) ( \cos(2 - 3x) - 3 \int \sin(2 - 3x)dx }} \\ { \sf{ = 2 {x}^(2) ( \cos(2 - 3x) - 3( - (1)/(3) \cos(2 - 3x) ) + c}} \\ { \sf{ = {4x}^(2) \cos(2 - 3x) + c}}`